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Academic grading in the United States
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Academic grading
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Academic grading in the United States commonly takes on the form of five letter grades. Traditionally, the grades are A, B, C, D, and F — A being the highest and F, short for failed, the lowest. Numeric to letter grade conversions generally vary from system to system and between disciplines and status. In some cases, grades can be numerical, from 1 to 4, in which 1 being the worst and 4 is the best.
Contents
[hide]
1 Numerical and Letter Grades
2 How grades are assigned to students
3 Standards-based grading
4 Rank-based grading
4.1 Weighted GPA
4.2 Six-point system
4.3 Eleven-point system
5 The 1-2-3-4 system
6 The E-S-N-U system
7 Alternative approaches to academic grading
8 Grade I or Y
9 Additional collegiate grades
9.1 Course audits
10 Standards of Academic Progress (SAP)
11 Grade points
12 References
Numerical and Letter Grades[edit]
The typical grades awarded for participation in a course are (from highest to lowest) A, B, C, D, and F. Variations on the traditional five-grade system allow for awarding A+, A, A- ; B+, B, & B-; C+, C, & C-; D+, D, & D-; and E and F. In primary and secondary schools, a D is usually the lowest passing grade, however, there are some schools that consider a C the lowest passing grade, so the general standard is that anything below a 60 or 70 is failing, depending on the grading scale. In college and universities, a D is considered to be an unsatisfactory passing grade. Students will usually still earn credit for the class if they get a D, but sometimes a C or better is required to count some major classes toward a degree, and sometimes a C or better is required to satisfy a prerequisite requirement for a class.
Below is the grading system found to be most commonly used in United States public high schools, according to the 2009 High School Transcript Study.[1] This is the most used grading system, however, there are some schools that use an edited version of the college system.
Letter Grade
Percentage
GPA
A 90%-100% 4.0
B 80%-89% 3.0
C 70%-79% 2.0
D 60%-69% 1.0
F 0%-59% 0.0
Below is a grading system used by four different colleges in the United States. Other schools use different systems, so this is by no means universal.[2][3][4][5][6]
Letter Grade
Percentage
GPA
A+ 97%+ 4.33/4.00 or 4.00/4.00
A 93%-96% 4.00/4.00
A- 90%-92% 3.67/4.00
B+ 87%-89% 3.33/4.00
B 83%-86% 3.00/4.00
B- 80%-82% 2.67/4.00
C+ 77%-79% 2.33/4.00
C 73%-76% 2.00/4.00
C- 70%-72% 1.67/4.00
D+ 67%-69% 1.33/4.00
D 63%-66% 1.00/4.00
D- 60%-62% 0.67/4.00
F 0%-59% 0.00/4.00
How grades are assigned to students[edit]
Most teachers construct an organized system for evaluating student work.
In a typical points-based system each question in every assignment is assigned a certain number of points. A simple homework question is usually worth five points and a lengthy project such as an essay is worth about 70 points. The points for a large project in turn may be further divided into smaller areas for evaluation (this is called a "rubric"): ten points for writing the correct length of an essay, five points for a well-written introduction, five points for spelling and grammar, ten points for reasoning, and so forth. The final grade for the course is calculated as a percentage of points earned out of points possible.
In a percentage-based system, each assignment regardless of size, type, or complexity, is given a percentage score: nine correct answers out of ten is a score of 90%. The overall grade for the class is then typically weighted so that the final grade represents a stated proportion of different types of work. For example, daily homework may be counted as 50% of the final grade, chapter quizzes may count for 20%, the comprehensive final exam may count for 20%, and a major project may count for the remaining 10%.
In elementary school, grades may represent rewards from teachers "for being friendly, prepared, compliant, a good school citizen, well organized and hard-working" rather than mastering the subject material.[7] Schools in the United States have been accused of using academic grades to penalize students for being bored, uncooperative or for talking out of turn. Usually, this behavior leads to poor or non-existent studying habits which most likely are to blame for their grades.[8] Also some teachers use Self- and Peer-Assessment to evaluate some of a student's work.
Standards-based grading[edit]
With the adoption of standards-based education, most states have created examinations in which students are compared to a standard of what educators, businesspeople, parents, and other stakeholders have determined to be what every student should know and be able to do. Students are graded as exceeding, meeting, or falling below the standard. The advantage is that students are not compared against each other, and all have the opportunity to pass the standard. However, the standard is typically set at a level that is substantially higher than previous achievement, so that a relatively high percentage of students fail at least some part of the standards in the first year, including an especially high percentage non-college bound students.
As an instrument of systemic reform, the tests are targeted to items and skills not currently in the curriculum[citation needed] to promote adoption of methods such as constructivist mathematics, inquiry-based science, and problem solving.
Grades can be enhanced by extra credits, awarded where students undertake optional work, additional to their compulsory school work.[9]
For an example of standard-based grading, see The 1-2-3-4 System, below.
Rank-based grading[edit]
Grading on a curve is any system wherein the group performance is used to moderate evaluation; it need not be strictly or purely rank-based.
In the most extreme form, students are ranked and grades are assigned according to a student's rank, placing students in direct competition with one another.
Ranking system that evenly distributes grades
Grade
percentage of students
receiving grade
A top 20%
B next 20%
C middle 20%
D next 20%
F bottom 20%
One model uses percentages derived from a normal distribution model of educational performance. The top grade, A, is given here for performance that exceeds the mean by +1.5 standard deviations, a B for performance between +0.5 and +1.5 standard deviations above the mean, and so on.[10] Regardless of the absolute performance of the students, the best score in the group receives a top grade, and the worst score receives a failing grade.
Rank-based grading is popular among some American educators, usually under the euphemism of grade rationing. The arguments for grade-rationing[according to whom?] are that:
Grade inflation, which is a serious problem in education [11] in which more students receive high grades, is mathematically impossible in a rank-based system, because the teacher cannot rate all students as being better than all of the other students: someone must be ranked better, and someone must be ranked worse. Historical measures of performance in the subject matter may no longer apply, as human knowledge has increased substantially over time. Rank-based grading compares current students to each other, rather than to a standard that may have been set decades before.
Rank-based grading may push some students to their greatest performance potential by appealing to their competitive instincts.
Rank-based grading shows how the student compares to other students, who all had the same instructor with the same lessons and homework during the same time period. If grades are meant to represent the student's relative ability to learn, rather than to certify that the student knows and can do certain things, then rank-based grading shows clear superiority in methodology to non-curved methods of grading. However, if the purpose of grading is purely to indicate abilities learned, then a non-ranked system is more appropriate.
As many corporations used rank-based evaluation measures, sometimes even related to termination (see: rank and yank) such grading prepares students for the corporate world. By limiting success and recognition to the top-performing students, the grading system becomes a relevant measure of student performance in relation to their peers. In this way, rank-based grading prevents the illusion that students are competitive in areas in which they are actually only competent.
The arguments against rank-based grading are similar:
Rank-based grading only measures performance relative to a given group, but not the real achievements of a given student. A student with moderate skills could be the best of a bad group, or the worst of a good group. For example, in a generally good class the pressure to assign grades along the curve would produce an artificial 7% of F students, although all students actually performed quite well. This also works the other way round: in a class with generally bad performance, the students whose performances are not totally bad would be singled out to form an artificial group of A-students, although in another context they would never get these grades.
There is no actual evidence that a given group really performs along the normal curve. The distribution may not match the pattern at all.
Rank-based grades become meaningless when taken out of the context of a given class or school. To understand what a rank-based grade indicates, it is necessary to understand the overall performance of the entire group on an absolute scale.
Numerical values are applied to grades as follows:
A = 4
B = 3
C = 2
D = 1
E/F = 0
This allows grades to be easily averaged. Additionally, many schools add .33 for a + grade and subtract .33 for a − grade. Thus, a B+ yields a 3.33 whereas an A– yields a 3.67.[12] A+s, if given, are usually assigned a value of 4.0 (equivalent to an A) due to the common assumption that a 4.00 is the best possible grade-point average, although 4.33 is awarded at some institutions. In some places, .25 or .3 instead of .33 is added for a + grade and subtracted for a − grade. Other institutions maintain a mid grade and award .5 for the grade. For example, an AB would receive a 3.5 grade point and a BC would receive a 2.5 grade point.
The industry standard for graduation from undergraduate institutions is a minimum 2.0 average. Most graduate schools have required a 3.0 grade point average since 1975 (the transition began two decades earlier), but some schools still have 2.75 as their pass standard. Some doctoral programs do not have a formal pass standard, but it is unlikely that they would retain a student who is doing work below 'B' quality.
Most American law schools require no more than a 2.0 grade point average to qualify for the professional doctorate in law. This is because law school grades are usually based on a strict bell curve system which typically results in the failure of 10-30% of first year students. A few law schools require 2.3 or 2.5 for post-doctoral degrees, such as the American LL.M. or S.J.D. degrees. Regular graduate schools have commonly eliminated the D grade because anything below a C is considered failing.
Weighted GPA[edit]
Some high schools, to reflect the varying skill required for different level courses and to discourage students from selecting courses that are considered a source of easy 'A's, will give higher numerical grades for difficult courses, often referred to as a weighted GPA. For example, two common conversion systems used in honors and advanced placement courses are:
A = 5.0 or 4.5
B = 4 or 3.5
C = 3 or 2.5
D = 2 or 1.5
E/F = 0[13]
Denver Public Schools uses a different system in honors and AP courses to get weighted GPA values; the scale is as follows: A = 5.2 A- = 4.77 B+ = 4.33 B = 3.9 B- = 3.47 C+ = 3.0 C = 2.6 C- = 2.17 D = 1.3 E = 0.0
Another policy commonly used by 4.0-scale schools is to mimic the eleven-point weighted scale (see below) by adding a .33 (one third of a letter grade) to an honors or advanced placement class. (For example, a B in a regular class would be a 3.0, but in an honors or AP class it would become a B+, or 3.33).
Sometimes the 5-based weighting scale is used for AP courses and the 4.6-based scale for honors courses, but often a school will choose one system and apply it universally to all advanced courses. A small number of high schools use a 5-point scale for Honors courses, a 6-point scale for AP courses, and/or a 3-point scale for courses of below average difficulty.
Although weighting GPAs is a widespread practice in the United States, there is little research into whether weighted GPAs are better than unweighted GPAs. In one study, weighted GPAs were not suitable for predicting any college outcomes, but unweighted GPA were strong predictors of college GPA.[14] However, standardized test scores were better predictors than either type of GPA.[14]
Six-point system[edit]
At least one boarding school in the United States uses the six-point system, Phillips Academy at Andover.[15] While there are approximate equivalents to the 100 point system, grades are most often described as follows:
6 - Outstanding (High Honor)
5 - Superior (Honor)
4 - Good
3 - Satisfactory
2 - Low Pass, but certifying
1 - Failure
0 - Low Failure
Though described above, grades below a 3 are rarely dispensed in practice.[15]
Eleven-point system[edit]
Phillips Exeter Academy and a few other high schools in the United States use an eleven-point system. Numerical values are applied to grades as follows:
A = 11
A− = 10
B+ = 9
B = 8
B− = 7
C+ = 6
C = 5
C− = 4
D+ = 3
D = 2
D− = 1
E = 0
Very few American high schools use a twelve-point system, which differs from the above only in using the grade A+, to which the value 12.0 is applied.
The 1-2-3-4 system[edit]
Some school districts use a 1-2-3-4 rating system for grades at the Elementary (K-5) level, notably many California school districts including The Los Angeles Unified School District (LAUSD) who switched with the class of 2000. The four-point scale more clearly indicates proficiency levels in core subjects by segmenting students who are proficient (4&3) and ready to advance, from those who are not meeting all required standards (2&1) and should not advance. Most notably this removes the "C" rating which did not clearly partition students who should advance from those who should not.[16]
Percentage
Grade/Mark
Assigned Meaning
Numeric Grade
90-100 45 Exemplary 4.0
80-89 36 Accomplished 3.0
70-79 21 Developing 2.0
0-69 05 Beginning 1.0
The E-S-N-U system[edit]
At one time (until roughly the mid-20th Century), the most popular grading system in the United States used four letters, which ranked in descending order:
E (Excellent)
S (Satisfactory)
N (needs improvement; "NI" was also used interchangeably)
U (unsatisfactory)
This system has largely been replaced by the five-point system discussed above, but is still encountered quite often at the elementary school level, particularly in kindergarten and Grades 1 through 3 (these levels comprising the lower division of primary school). It is also occasionally used at schools for older children, including high schools, especially in the issuance of conduct or citizenship grades.
There are a few variations to this system, including the use of an O (for "outstanding") grade, which is even higher than the E; the use of an O instead of the E; a G (for "good") placed between the E and the S; the use of a G (again for "good") instead of the E; and the lack of a U grade. In this version, E stands for "exemplary" and P proficient, with AE and AP for work that approaches the E and P levels. "Credit" is equivalent to the D level and "No Credit" is equivalent to F. [1]
The use of M (for "mediocre") in place of the N and I (for "insufficient") in place of the U was used in the midwest, and included the F.
E (Excellent)
S (Satisfactory)
M (Mediocre)
I (Insufficient)
F (Failure)
The S grade may be so modified with an S+ or S−, but otherwise plus and minus are rarely used.
A similar system is used to rank practical work in the certain science department of Oxford University; however only with the grades S (Satisfactory) S+ (more than satisfactory, and may be used in the allocation of degree grades) and NS (Not Satisfactory).
Alternative approaches to academic grading[edit]
Alternatives to letter-grading assessments have been tried in some schools, but still remain a marginal approach due to the heavy emphasis and history of letter grading.[17] Researchers including Carol Dweck, Carole Ames, Ruth Butler, and John Nicholls suggest that students' focus on their achievement takes away from their actual learning.[citation needed]
A number of liberal-arts colleges in the U.S. either do not issue grades at all (such as Alverno College, Antioch College, Bennington College, The Evergreen State College, New College of Florida, and Hampshire College), de-emphasize them (St. John's College, Reed College, Sarah Lawrence College, Prescott College, College of the Atlantic), or do not calculate grade point averages (Brown University[18]). In all cases, the rationale is that grades alone do not provide a clear picture of academic aptitude or of potential for success and that learning, not achieving the highest score, should be the goal of a liberal education as is the case in graduate programs.[citation needed] In many cases, narrative evaluations are used as an alternative measurement system. Saint Ann's School in Brooklyn is one of several secondary schools to eschew grades in favor of narrative reports, while still managing to be the number one high school in the country for having the highest percentage of graduating seniors enroll in Ivy League and several other highly selective colleges.[19]
Grade I or Y[edit]
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The letter grades 'I' and 'Y' are temporary grades, representing "Incomplete" and "Year", respectively. The distinction between them is so slight that most institutions only use one of them, not both.
A 'Y' grade indicates the course, or the time allowed to complete it extends beyond the temporal bounds of a single term to encompass an entire academic or calendar year, or longer. Conversely, an 'I' grade indicates the student was satisfactorily completing a course when something happened that prevented timely completion, usually illness or injury.
The length of time allowed to complete the work for an 'I' grade varies from a deadline in the first few weeks following the end of term to a full calendar year, the latter being more usual. For some courses (such as independent studies or thesis/dissertation credits), or in some situations (such as subsequent non-enrollment), the time allowed for completion may be indefinite. Some institutions will convert the 'I' grade to an 'E' or 'F' (failing) grade if the student does not complete the course by the end of the time period; others simply make the 'I' grade permanent at that time; and a few institutions retroactively withdraw the student from the course, changing the 'I' to a 'W' grade. Most institutions allow students to apply for an extension of the completion time, upon presentation of special circumstances.
Policies on 'Y' grades are similar or identical to those for 'I' grades, but because of the historical difference in meaning ("Year" instead of "Incomplete"), it is rare for 'Y' grades to be converted to a failing grade if a student does not finish the course, but conversion to 'W' may be done.
Additional collegiate grades[edit]
FN = Failure for Non-Attendance
W = Withdrawal
WP = Withdrawal (had a passing grade at the time of withdrawal)
WF = Withdrawal (had a failing grade at the time of withdrawal)
UW = Unofficial Withdrawal
X = Audit/Exemption
NR = Not Reported by Instructor
E = Excellent
The FN grade indicates that a student has failed a course due to non-attendance. It is calculated as an "F" in the student’s grade point average. For students receiving financial aid, failure for non-attendance may require the student to refund to the College all or part of his or her aid. The FN grade will be assigned by the faculty member at any time following the final withdrawal date for the course. Students who are in a failing status because of non-attendance but return to the course prior to the withdrawal date may elect to withdraw from the course.
A grade of "W" indicates that a student has elected to withdrawal from a course prior to the course's withdraw deadline. It is not calculated in the student’s grade point average, which would keep the student from facing possible academic disciplinary action if he or she was to fall below the required Standards of Academic Progress (SAP). For students receiving financial aid, a grade of "W" may require the student to refund to the College all or part of his or her aid. Some schools indicate whether the student was passing or failing the course at the time of withdrawal by placing WP or WF grades on the transcript; policies vary as to whether a WF counts as an unsatisfactory grade when determining if a student is in good standing.
Standards for Academic Progress in Florida, for example, require a student to maintain a grade point average of 2.00 on the 4.00 scale. The student must also successfully complete 67% of the courses attempted, which includes previous failures, re-takes, and withdrawals. Additionally, a student may not attempt a course more than three (3) times.
Course audits[edit]
Students may elect to audit a college credit course or workforce credit course by completing the audit form. Students may not change from credit to audit or from audit to credit after the drop deadline. A grade of "X" will be assigned for all courses taken in audit status.
No credit will be awarded and fees for college credit courses taken on an audit basis are the same as those taken on a college credit or workforce credit basis.
Courses taken for audit do not count as hours enrolled in the following areas: veteran certification, financial aid awards, Social Security certification, international student enrollment requirements or early admission program enrollment requirements.
Standards of Academic Progress (SAP)[edit]
Standards of Academic Progress are the standards set by the school, state, Board of Education, or other agency which are required of students to adhere to in order to continue to attend classes. A student who falls below the SAP may have disciplinary action taken against him or her or denial of financial aid until the student has met the required SAP. In Florida, Standards of Academic Progress require a student to maintain a grade point average of 2.00 or above on the 4.00 numeric grading scale. The student must also finish 67% of the courses attempted, which includes previous failures, re-takes, and withdrawals. Additionally, a student may not attempt a course more than three (3) times.
Grade points[edit]
To evaluate the scholastic standing of students, the following points are assigned to grades.
A = four grade points per semester hour
B = three grade points per semester hour
C = two grade points per semester hour
D = one grade point per semester hour
E/F = zero grade points per semester hour
EN = zero grade points per semester hour
Students’ scholastic standing or grade point average is obtained by dividing the total number of grade points earned by the total number of semester hours attempted for which the foregoing grades have been assigned. Grades of "I," "W," "NR," and "X" are not used in the computation of grade point average. Grades earned in college preparatory classes do not count in the computation of the grade point average.
References[edit]
Jump up ^ "How is Grade Average Calculated?". National Center for Educational Statistics. U.S. Department of Education. Retrieved 14 February 2015.
Jump up ^ "Dutchess Community College Grading System". Dutchess Community College. Retrieved 14 February 2015.
Jump up ^ "Grading Practices | Academic Policies | Policies | Academic Catalog | | Kalamazoo College". www.kzoo.edu. Retrieved 2017-12-04.
Jump up ^ "Carleton College Grading System". Carleton College. Retrieved 14 February 2015.
Jump up ^ "Wellesley College Grading System and Policies". Wellesley College. Retrieved 14 February 2015.
Jump up ^ "InstructorLink Teaching in Centers Grading Chart and Grade Descriptions - UC Berkeley Extension". instructorlink.berkeley.edu. Retrieved 2017-11-30.
Jump up ^ Tyre, Peggy (27 November 2010). "A's for Good Behavior". The New York Times.
Jump up ^ McKeown, Joe (13 July 2015). "The Problem with Standards-Based Grading".
Jump up ^ Lucas, Sandra Goss; Bernstein, Douglas A. (2004). Teaching Psychology. p. 36. ISBN 1-4051-5150-1.
Jump up ^ Robert E. Slavin, Educational Psychology: Theory into Practice, (Englewood Cliffs, N.J., Prentice-Hall, 1986, p.556-57
Jump up ^ Valerie Strauss, "Why grade inflation (even at Harvard) is a big problem," The Washington Post, December 20, 2013. https://www.washingtonpost.com/news/answer-sheet/wp/2013/12/20/why-grade-inflation-even-at-harvard-is-a-big-problem/
Jump up ^ http://studentspreunited.com/page/GPA4pointcalculator
Jump up ^ "High School GPA Calculator", Studentspreunited.com. Accessed 19 November 2011.
^ Jump up to: a b Warne, R. T., *Nagaishi, C., *Slade, M. K., Hermesmeyer, P., & Peck, E. K. (2014). Comparing weighted and unweighted grade point averages (GPAs) in predicting college success in diverse and low-income college students. NASSP Bulletin, 98, 261-279. doi:10.1177/0192636514565171
^ Jump up to: a b https://www.andover.edu/Academics/CollegeCounseling/Documents/PhillipsAcademySchoolProfile2013-2014.pdf
Jump up ^ Kondo, Annette. "LAUSD Switches From Letter to Number Grades". Los Angeles Times.
Jump up ^ Kohn, Alfie (1999). "From Degrading to De-grading". High School Magazine. |access-date= requires |url= (help)
Jump up ^ Brown University (March 2004). "Statement on Grade Point Average" (PDF). Brown University. Retrieved 7 April 2015.
Jump up ^ April 2, 2004 Wall Street Journal, Cover Story (Personal Journal)
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Equilateral triangle
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"Equilateral" redirects here. For other uses, see Equilateral (disambiguation).
Equilateral triangle
Type
Regular polygon
Edges and vertices
3
Schläfli symbol
{3}
Coxeter diagram
Symmetry group
D3
Area
3 4 a 2 {\displaystyle {\tfrac {\sqrt {3}}{4}}a^{2}}
Internal angle (degrees)
60°
In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, equilateral triangles are also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. They are regular polygons, and can therefore also be referred to as regular triangles.
Contents
[hide]
1 Principal properties
2 Characterizations
2.1 Sides
2.2 Semiperimeter
2.3 Angles
2.4 Area
2.5 Circumradius, inradius and exradii
2.6 Equal cevians
2.7 Coincident triangle centers
2.8 Six triangles formed by partitioning by the medians
2.9 Points in the plane
3 Notable theorems
4 Other properties
5 Geometric construction
6 Derivation of area formula
6.1 Using the Pythagorean theorem
6.2 Using trigonometry
7 In culture and society
8 See also
9 References
10 External links
Principal properties[edit]
An equilateral triangle. It has equal sides (a=b=c), equal angles ( α = β = γ {\displaystyle \alpha =\beta =\gamma } ), and equal altitudes (ha=hb=hc).
Denoting the common length of the sides of the equilateral triangle as a, we can determine using the Pythagorean theorem that:
The area is A = 3 4 a 2 {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}}
The perimeter is p = 3 a {\displaystyle p=3a\,\!}
The radius of the circumscribed circle is R = a 3 {\displaystyle R={\frac {a}{\sqrt {3}}}}
The radius of the inscribed circle is r = 3 6 a {\displaystyle r={\frac {\sqrt {3}}{6}}a} or r = R 2 {\displaystyle r={\frac {R}{2}}}
The geometric center of the triangle is the center of the circumscribed and inscribed circles
And the altitude (height) from any side is h = 3 2 a {\displaystyle h={\frac {\sqrt {3}}{2}}a} .
Denoting the radius of the circumscribed circle as R, we can determine using trigonometry that:
The area of the triangle is A = 3 3 4 R 2 {\displaystyle \mathrm {A} ={\frac {3{\sqrt {3}}}{4}}R^{2}}
Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
The area is A = h 2 3 {\displaystyle A={\frac {h^{2}}{\sqrt {3}}}}
The height of the center from each side, or apothem, is h 3 {\displaystyle {\frac {h}{3}}}
The radius of the circle circumscribing the three vertices is R = 2 h 3 {\displaystyle R={\frac {2h}{3}}}
The radius of the inscribed circle is r = h 3 {\displaystyle r={\frac {h}{3}}}
In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors and the medians to each side coincide.
Characterizations[edit]
A triangle ABC that has the sides a, b, c, semiperimeter s, area T, exradii ra, rb, rc (tangent to a, b, c respectively), and where R and r are the radii of the circumcircle and incircle respectively, is equilateral if and only if any one of the statements in the following nine categories is true. Thus these are properties that are unique to equilateral triangles, and knowing that any one of them is true directly implies that we have an equilateral triangle.
Sides[edit]
a = b = c {\displaystyle \displaystyle a=b=c}
1 a + 1 b + 1 c = 25 R r − 2 r 2 4 R r {\displaystyle \displaystyle {\frac {1}{a}}+{\frac {1}{b}}+{\frac {1}{c}}={\frac {\sqrt {25Rr-2r^{2}}}{4Rr}}} [1]
Semiperimeter[edit]
s = 2 R + ( 3 3 − 4 ) r (Blundon) {\displaystyle \displaystyle s=2R+(3{\sqrt {3}}-4)r\quad {\text{(Blundon)}}} [2]
s 2 = 3 r 2 + 12 R r {\displaystyle \displaystyle s^{2}=3r^{2}+12Rr} [3]
s 2 = 3 3 T {\displaystyle \displaystyle s^{2}=3{\sqrt {3}}T} [4]
s = 3 3 r {\displaystyle \displaystyle s=3{\sqrt {3}}r}
s = 3 3 2 R {\displaystyle \displaystyle s={\frac {3{\sqrt {3}}}{2}}R}
Angles[edit]
A = B = C = 60 ∘ {\displaystyle \displaystyle A=B=C=60^{\circ }}
cos  A + cos  B + cos  C = 3 2 {\displaystyle \displaystyle \cos {A}+\cos {B}+\cos {C}={\frac {3}{2}}}
sin  A 2 sin  B 2 sin  C 2 = 1 8 {\displaystyle \displaystyle \sin {\frac {A}{2}}\sin {\frac {B}{2}}\sin {\frac {C}{2}}={\frac {1}{8}}} [5]
Area[edit]
T = a 2 + b 2 + c 2 4 3 {\displaystyle \displaystyle T={\frac {a^{2}+b^{2}+c^{2}}{4{\sqrt {3}}}}\quad } (Weitzenböck) [6]
T = 3 4 ( a b c ) 2 3 {\displaystyle \displaystyle T={\frac {\sqrt {3}}{4}}(abc)^{^{\frac {2}{3}}}} [4]
Circumradius, inradius and exradii[edit]
R = 2 r (Chapple-Euler) {\displaystyle \displaystyle R=2r\quad {\text{(Chapple-Euler)}}} [7]
9 R 2 = a 2 + b 2 + c 2 {\displaystyle \displaystyle 9R^{2}=a^{2}+b^{2}+c^{2}} [7]
r = r a + r b + r c 9 {\displaystyle \displaystyle r={\frac {r_{a}+r_{b}+r_{c}}{9}}} [5]
r a = r b = r c {\displaystyle \displaystyle r_{a}=r_{b}=r_{c}}
Equal cevians[edit]
Three kinds of cevians are equal for (and only for) equilateral triangles:[8]
The three altitudes have equal lengths.
The three medians have equal lengths.
The three angle bisectors have equal lengths.
Coincident triangle centers[edit]
Every triangle center of an equilateral triangle coincides with its centroid, which implies that the equilateral triangle is the only triangle with no Euler line connecting some of the centers. For some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular:
A triangle is equilateral if any two of the circumcenter, incenter, centroid, or orthocenter coincide.[9]:p.37
It is also equilateral if its circumcenter coincides with the Nagel point, or if its incenter coincides with its nine-point center.[7]
Six triangles formed by partitioning by the medians[edit]
For any triangle, the three medians partition the triangle into six smaller triangles.
A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.[10]:Theorem 1
A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.[10]:Corollary 7
Points in the plane[edit]
A triangle is equilateral if and only if, for every point P in the plane, with distances p, q, and r to the triangle's sides and distances x, y, and z to its vertices,[11]:p.178,#235.4
4 ( p 2 + q 2 + r 2 ) ≥ x 2 + y 2 + z 2 . {\displaystyle 4(p^{2}+q^{2}+r^{2})\geq x^{2}+y^{2}+z^{2}.}
Notable theorems[edit]
Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.
A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.[12]
Viviani's theorem states that, for any interior point P in an equilateral triangle with distances d, e, and f from the sides and altitude h,
d + e + f = h , {\displaystyle d+e+f=h,}
independent of the location of P.[13]
Pompeiu's theorem states that, if P is an arbitrary point in an equilateral triangle ABC, then there exists a triangle with sides of lengths PA, PB, and PC. That is, PA, PB, and PC satisfy the triangle inequality that any two of them sum to at least as great as the third.
Other properties[edit]
By Euler's inequality, the equilateral triangle has the smallest ratio R/r of the circumradius to the inradius of any triangle: specifically, R/r = 2.[14]:p.198
The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.[15]
The ratio of the area of the incircle to the area of an equilateral triangle, π 3 3 {\displaystyle {\frac {\pi }{3{\sqrt {3}}}}} , is larger than that of any non-equilateral triangle.[16]:Theorem 4.1
The ratio of the area to the square of the perimeter of an equilateral triangle, 1 12 3 , {\displaystyle {\frac {1}{12{\sqrt {3}}}},} is larger than that for any other triangle.[12]
If a segment splits an equilateral triangle into two regions with equal perimeters and with areas A1 and A2 , then[11]:p.151,#J26
7 9 ≤ A 1 A 2 ≤ 9 7 . {\displaystyle {\frac {7}{9}}\leq {\frac {A_{1}}{A_{2}}}\leq {\frac {9}{7}}.}
If a triangle is placed in the complex plane with complex vertices z1, z2, and z3, then for either non-real cube root ω {\displaystyle \omega } of 1 the triangle is equilateral if and only if[17]:Lemma 2
z 1 + ω z 2 + ω 2 z 3 = 0. {\displaystyle z_{1}+\omega z_{2}+\omega ^{2}z_{3}=0.}
Given a point P in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when P is the centroid. In no other triangle is there a point for which this ratio is as small as 2.[18] This is the Erdős–Mordell inequality; a stronger variant of it is Barrow's inequality, which replaces the perpendicular distances to the sides with the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides (A, B, and C being the vertices).
For any point P in the plane, with distances p, q, and t from the vertices A, B, and C respectively,[19]
3 ( p 4 + q 4 + t 4 + a 4 ) = ( p 2 + q 2 + t 2 + a 2 ) 2 . {\displaystyle \displaystyle 3(p^{4}+q^{4}+t^{4}+a^{4})=(p^{2}+q^{2}+t^{2}+a^{2})^{2}.}
For any point P on the inscribed circle of an equilateral triangle, with distances p, q, and t from the vertices,[19]
4 ( p 2 + q 2 + t 2 ) = 5 a 2 {\displaystyle \displaystyle 4(p^{2}+q^{2}+t^{2})=5a^{2}}
and
16 ( p 4 + q 4 + t 4 ) = 11 a 4 . {\displaystyle \displaystyle 16(p^{4}+q^{4}+t^{4})=11a^{4}.}
For any point P on the minor arc BC of the circumcircle, with distances p, q, and t from A, B, and C respectively,[13]:170[19]
p = q + t {\displaystyle \displaystyle p=q+t}
and
q 2 + q t + t 2 = a 2 ; {\displaystyle \displaystyle q^{2}+qt+t^{2}=a^{2};}
moreover, if point D on side BC divides PA into segments PD and DA with DA having length z and PD having length y, then[13]:172
z = t 2 + t q + q 2 t + q , {\displaystyle z={\frac {t^{2}+tq+q^{2}}{t+q}},}
which also equals t 3 − q 3 t 2 − q 2 {\displaystyle {\tfrac {t^{3}-q^{3}}{t^{2}-q^{2}}}} if t ≠ q; and
1 q + 1 t = 1 y , {\displaystyle {\frac {1}{q}}+{\frac {1}{t}}={\frac {1}{y}},}
which is the optic equation.
There are numerous triangle inequalities that hold with equality if and only if the triangle is equilateral.
An equilateral triangle is the most symmetrical triangle, having 3 lines of reflection and rotational symmetry of order 3 about its center. Its symmetry group is the dihedral group of order 6 D3.
Equilateral triangles are the only triangles whose Steiner inellipse is a circle (specifically, it is the incircle).
The integer-sided equilateral triangle is the only triangle with integer sides and three rational angles as measured in degrees.[20]
A regular tetrahedron is made of four equilateral triangles.
Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform polyhedra. Three of the five Platonic solids are composed of equilateral triangles. In particular, the regular tetrahedron has four equilateral triangles for faces and can be considered the three-dimensional analogue of the shape. The plane can be tiled using equilateral triangles giving the triangular tiling.
Geometric construction[edit]
Construction of equilateral triangle with compass and straightedge
An equilateral triangle is easily constructed using a compass and straightedge, as 3 is a Fermat prime. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment
An alternative method is to draw a circle with radius r, place the point of the compass on the circle and draw another circle with the same radius. The two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.
In both methods a by-product is the formation of vesica piscis.
The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of Euclid's Elements.
Derivation of area formula[edit]
The area formula A = 3 4 a 2 {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}} in terms of side length a can be derived directly using the Pythagorean theorem or using trigonometry.
Using the Pythagorean theorem[edit]
The area of a triangle is half of one side a times the height h from that side:
A = 1 2 a h . {\displaystyle A={\frac {1}{2}}ah.}
An equilateral triangle with a side of 2 has a height of √3 as the sine of 60° is √3/2.
The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base a, and the hypotenuse is the side a of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem
( a 2 ) 2 + h 2 = a 2 {\displaystyle \left({\frac {a}{2}}\right)^{2}+h^{2}=a^{2}}
so that
h = 3 2 a . {\displaystyle h={\frac {\sqrt {3}}{2}}a.}
Substituting h into the area formula (1/2)ah gives the area formula for the equilateral triangle:
A = 3 4 a 2 . {\displaystyle A={\frac {\sqrt {3}}{4}}a^{2}.}
Using trigonometry[edit]
Using trigonometry, the area of a triangle with any two sides a and b, and an angle C between them is
A = 1 2 a b sin  C . {\displaystyle A={\frac {1}{2}}ab\sin C.}
Each angle of an equilateral triangle is 60°, so
A = 1 2 a b sin  60 ∘ . {\displaystyle A={\frac {1}{2}}ab\sin 60^{\circ }.}
The sine of 60° is 3 2 {\displaystyle {\tfrac {\sqrt {3}}{2}}} . Thus
A = 1 2 a b × 3 2 = 3 4 a b = 3 4 a 2 {\displaystyle A={\frac {1}{2}}ab\times {\frac {\sqrt {3}}{2}}={\frac {\sqrt {3}}{4}}ab={\frac {\sqrt {3}}{4}}a^{2}}
since all sides of an equilateral triangle are equal.
In culture and society[edit]
Equilateral triangles have frequently appeared in man made constructions:
Some archaeological sites have equilateral triangles as part of their construction, for example Lepenski Vir in Serbia.
The shape also occurs in modern architecture such as Randhurst Mall and the Jefferson National Expansion Memorial.
The Flag of Nicaragua, the Flag of the Philippines, the Seal of the President of the Philippines, and the Flag of Junqueirópolis contain equilateral triangles.
It is a shape of a variety of road signs, including the Yield sign.
The fraternity Tau Kappa Epsilon uses the equilateral triangle as its primary symbol.
See also[edit]
Almost-equilateral Heronian triangle
Isosceles triangle
Right triangle
Trigonometry
Ternary plot
References[edit]
Jump up ^ Bencze, Mihály; Wu, Hui-Hua; Wu, Shan-He (2008). "An equivalent form of fundamental triangle inequality and its applications" (PDF). Research Group in Mathematical Inequalities and Applications. 11 (1).
Jump up ^ Dospinescu, G.; Lascu, M.; Pohoata, C.; Letiva, M. (2008). "An elementary proof of Blundon's inequality" (PDF). Journal of inequalities in pure and applied mathematics. 9 (4).
Jump up ^ Blundon, W. J. (1963). "On Certain Polynomials Associated with the Triangle". Mathematics Magazine. 36 (4): 247–248. doi:10.2307/2687913.
^ Jump up to: a b Alsina, Claudi; Nelsen, Roger B. (2009). When less is more. Visualizing basic inequalities. Mathematical Association of America. pp. 71, 155.
^ Jump up to: a b Pohoata, Cosmin (2010). "A new proof of Euler's inradius - circumradius inequality" (PDF). Gazeta Matematica Seria B (3): 121–123.
Jump up ^ McLeman, Cam; Ismail, Andrei. "Weizenbock's inequality". PlanetMath. Archived from the original on 2012-02-18.
^ Jump up to: a b c Andreescu, Titu; Andrica, Dorian (2006). Complex Numbers from A to...Z. Birkhäuser. pp. 70, 113–115.
Jump up ^ Owen, Byer; Felix, Lazebnik; Deirdre, Smeltzer (2010). Methods for Euclidean Geometry. Mathematical Association of America. pp. 36, 39.
Jump up ^ Yiu, Paul (1998). "Notes on Euclidean Geometry" (PDF). [permanent dead link]
^ Jump up to: a b Cerin, Zvonko (2004). "The vertex-midpoint-centroid triangles" (PDF). Forum Geometricorum. 4: 97–109.
^ Jump up to: a b "Inequalities proposed in "Crux Mathematicorum"" (PDF).
^ Jump up to: a b Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.
^ Jump up to: a b c Posamentier, Alfred S.; Salkind, Charles T. (1996). Challenging Problems in Geometry. Dover Publ.
Jump up ^ Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities" (PDF). Forum Geometricorum. 12: 197–209.
Jump up ^ Dörrie, Heinrich (1965). 100 Great Problems of Elementary Mathematics. Dover Publ. pp. 379–380.
Jump up ^ Minda, D.; Phelps, S. (2008). "Triangles, ellipses, and cubic polynomials". American Mathematical Monthly. 115 (October): 679–689. JSTOR 27642581.
Jump up ^ Dao, Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers" (PDF). Forum Geometricorum. 15: 105–114.
Jump up ^ Lee, Hojoo (2001). "Another proof of the Erdős–Mordell Theorem" (PDF). Forum Geometricorum. 1: 7–8.
^ Jump up to: a b c De, Prithwijit (2008). "Curious properties of the circumcircle and incircle of an equilateral triangle". Mathematical Spectrum. 41 (1): 32–35.
Jump up ^ Conway, J. H., and Guy, R. K., "The only rational triangle", in The Book of Numbers, 1996, Springer-Verlag, pp. 201 and 228–239.
External links[edit]
Weisstein, Eric W. "Equilateral Triangle". MathWorld.
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Fundamental convex regular and uniform polytopes in dimensions 2–10
Family
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon
Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron
Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope
5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope
5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope
6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope
7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope
8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope
9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope
10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope
n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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value: {\displaystyle \displaystyle A=B=C=60^{\circ }}

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