Triangles: Types, Angles, Area, Centers and Congruence

What is a triangle?

A triangle is a polygon with exactly three sides, three vertices (corners) and three interior angles. It is the simplest polygon, and every other polygon can be cut into triangles: a diagonal splits a quadrilateral into two. The Indonesian word is segitiga, "three-corner".

We name a triangle by its vertices, ABC. The side opposite the vertex A is written a (it is BC), the side opposite B is b (it is CA) and the side opposite C is c (it is AB). The interior angles are \angle A, \angle B, \angle C.

Any side can serve as the base. The height (or altitude) belonging to that base is the perpendicular distance from the opposite vertex to the line that holds the base. In an obtuse triangle this height falls outside the triangle, on the extension of the base.

A triangle is rigid: its three side lengths fix its shape completely, so it cannot be pushed out of shape without breaking a side. A four-sided frame can lean into a parallelogram, a triangle cannot, which is why bridges, roof trusses, bicycle frames and camera tripods are built from triangles.

Why do the angles of a triangle add up to 180°?

The three interior angles of every triangle add up to 180^\circ, because a line through one vertex parallel to the opposite side makes the three angles fit together along a straight line.

Draw the line through C parallel to AB. The two angles it makes with CA and CB are alternate angles with \angle A and \angle B, so they are equal to them. Now \angle A, \angle C and \angle B sit side by side on the straight line through C, and a straight angle is 180^\circ:

\angle A+\angle B+\angle C=180^\circ.

Three consequences follow at once:

• A triangle has at most one right angle or obtuse angle, since two of them would already use 180^\circ or more.
• Knowing two angles gives the third: 50^\circ and 60^\circ leave 70^\circ.
• The exterior angle at C, formed by extending AC past C (or BC), equals \angle A+\angle B, the sum of the two interior angles not next to it, because both it and \angle C are what is left of 180^\circ. With \angle A=50^\circ and \angle B=60^\circ the exterior angle at C is 110^\circ.

This is also where the 360^\circ of a quadrilateral comes from: two triangles. If three angles are in the ratio 1:2:3, call them x,2x,3x as in turning words into algebra: 6x=180, so x=30^\circ and the angles are 30^\circ,60^\circ,90^\circ, a right triangle.

What are the types of triangles?

Triangles are sorted in two independent ways: by their sides (equilateral, isosceles, scalene) and by their largest angle (acute, right, obtuse), so every triangle has one name from each list.

Two rules link sides and angles. Equal sides lie opposite equal angles, and a longer side lies opposite a larger angle. So the longest side of a triangle is opposite its largest angle, and in a right triangle that side, the hypotenuse, is opposite the right angle.

Not every pair of names can occur together. An equilateral triangle is always acute. An isosceles triangle can be acute (5,5,6), right (1,1,√2, the half of a square cut along its diagonal) or obtuse (5,5,8). A scalene triangle can be acute (4,5,6), right (3,4,5) or obtuse (2,3,4).

To find the angle type without any angle, look at the longest side c and compare c^2 with a^2+b^2: if c^2<a^2+b^2 the triangle is acute, if c^2=a^2+b^2 it is right, and if c^2>a^2+b^2 it is obtuse. For 2,3,4: 16>4+9=13, so it is obtuse. This is the Pythagorean theorem and its two extensions, and the next sections use it.

Can any three lengths form a triangle?

No: three lengths form a triangle only if each is shorter than the sum of the other two, which is the same as saying that the longest is shorter than the other two together. This is the triangle inequality, and it says that a straight path is shorter than a detour: going from A to B directly is shorter than going through C, so c<a+b.

Check 3,4,5: 5<3+4, so it is a triangle. Check 1,2,3: 3=1+2, so the short sides lie flat along the long one and the triangle is degenerate, a segment. Check 1,2,4: 4>1+2, the short sides cannot even reach each other.

The inequality also tells you how long the third side can be. If two sides are a and b, the third side c satisfies |a-b|<c<a+b. With sides 5 and 8 the third side lies strictly between 3 and 13, so a whole number can be 4,5,…,12, which is nine values.

Type three lengths below. The tool uses exact fractions: it tells you whether they form a triangle, whether it is acute, right or obtuse by comparing squares, and gives the area by Heron's formula (more on that soon), the radii of the two circles and the angles.

What is special about right triangles?

A right triangle has one 90^\circ angle; the side opposite it is the hypotenuse, the longest side, and the other two sides are the legs. Its other two angles are acute and add up to 90^\circ, since the three angles make 180^\circ.

The sides of a right triangle obey the Pythagorean theorem, a^2+b^2=c^2. That theorem has its own article, with proofs, whole-number triples and applications; the section on types above already uses it in reverse, to tell a right angle from three lengths.

Two right triangles are worth knowing by heart, because their sides come in fixed ratios:

• The 45^\circ–45^\circ–90^\circ triangle (half a square) has sides in the ratio 1:1:√2: with legs a the hypotenuse is a√2.
• The 30^\circ–60^\circ–90^\circ triangle (half an equilateral triangle) has sides in the ratio 1:√3:2: the short leg is half the hypotenuse and the long leg is the short leg times √3.

In a right triangle the two legs serve as base and height, so the area is simply \frac12ab, and the circumcenter is the midpoint of the hypotenuse.

How do you find the perimeter and area of a triangle?

The perimeter is a+b+c, and the area is half the base times the height; when the height is not known, use Heron's formula for three sides, \frac12ab\sin C for two sides and the angle between them, or the coordinate formula.

Why one half? Two copies of a triangle, one turned around, fit together into a parallelogram with the same base and height, whose area is bh as in the area of a quadrilateral. The triangle is half of it.

A worked example. The triangle with sides 13,14,15 has s=21, so the area is √(21·8·7·6)=√(7056)=84. Taking 14 as the base, the height is h=(2·84)/(14)=12. The inradius is r=(84)/(21)=4 and the circumradius is R=(abc)/(4K)=(13·14·15)/(4·84)=(65)/(8).

The coordinate formula for A(0,0), B(6,0), C(0,8) gives \frac12|0+6·8+0|=24, which is \frac12·6·8 for the right triangle. The sine formula for sides 7 and 10 with a 30^\circ angle between them gives \frac12·7·10·\sin30^\circ=17.5. An equilateral triangle of side 6 has area (√3)/(4)·36=9√3≈15.59.

A word problem: a plot of land is a triangle with a base of 24 m and a height of 15 m, and land costs 50 dollars per square meter. The area is \frac12·24·15=180 m², so the plot costs 180·50=9000 dollars.

What are the medians, altitudes and centers of a triangle?

A triangle has four classic centers, and each is the point where three special lines meet: the centroid (medians), the circumcenter (perpendicular bisectors), the orthocenter (altitudes) and the incenter (angle bisectors).

The centroid and the incenter are always inside the triangle. The circumcenter and orthocenter are inside an acute triangle, at the midpoint of the hypotenuse and at the right-angle vertex of a right triangle, and outside an obtuse one.

The Euler line. Euler showed in 1765 that O, G and H lie on one line, and that G is one third of the way from O to H: OG:GH=1:2. In an equilateral triangle all three coincide; the incenter lies on that line only for an isosceles triangle.

For the right triangle A(0,0), B(6,0), C(0,8) the circumcenter is the midpoint of the hypotenuse, O=(3,4), with R=5; the orthocenter is the right-angle vertex, H=(0,0); and the centroid is the average of the vertices, G=\left(2,\frac83\right). Check the line: O+\frac13(H-O)=(3-1, 4-\frac43)=\left(2,\frac83\right)=G. The inradius is r=(6+8-10)/(2)=2.

Move the vertices below and watch the four centers, the two circles and the Euler line. The three exact centers are fractions, because the formulas use only addition, multiplication and division.

How do you tell whether two triangles are congruent or similar?

Two triangles are congruent (the same shape and size) if three well-chosen measurements agree, and similar (the same shape, possibly a different size) if their angles agree or their sides are proportional.

Side-side-angle (SSA), where the angle is not between the sides, is not a test: the same a, b and A can make two different triangles, as the solver below shows. The only exception is the right-triangle case RHS. Angle-angle-angle (AAA) fixes the shape but not the size, so it only proves similarity.

If two triangles are similar with scale factor k, every length is multiplied by k, every angle is unchanged, and areas are multiplied by k^2. This is the idea behind measuring a tall object by its shadow: a person 1.8 m tall casts a shadow of 2.4 m while a tree casts 12 m at the same moment. The triangles are similar (the sun's angle is the same), so (h)/(12)=(1.8)/(2.4) and the tree is h=9 m tall. Setting up such an equation is the ratio and proportion idea.

How do you solve a triangle with the law of sines and cosines?

To solve a triangle means to find all three sides and all three angles from three measurements, using the law of cosines, c^2=a^2+b^2-2ab\cos C, and the law of sines, (a)/(\sin A)=(b)/(\sin B)=(c)/(\sin C).

The law of cosines is the Pythagorean theorem with a correction: at C=90^\circ the cosine is 0 and it becomes c^2=a^2+b^2. Its sign decides the type: \cos C>0 (acute), =0 (right) or <0 (obtuse), which is the test in the section on types. The law of sines says that each side is proportional to the sine of the opposite angle.

Examples. SSS with 7,8,9: \cos A=(64+81-49)/(2·8·9)=\frac23, so A≈48.19^\circ; likewise B≈58.41^\circ and C=180-48.19-58.41≈73.40^\circ. SAS with a=5, b=8 and C=60^\circ: c^2=25+64-2·5·8·\frac12=49, so c=7.

The ambiguous case. Take a=3, b=5 and A=30^\circ. The law of sines gives \sin B=(5\sin30^\circ)/(3)=\frac56≈0.8333, and two angles have that sine, B≈56.44^\circ and B≈123.56^\circ. Both leave a positive third angle (93.56^\circ or 26.44^\circ), so two triangles fit: with c≈5.99 and with c≈2.67. In general, with A acute, side a too short (a<b\sin A) gives no triangle, a=b\sin A gives one right triangle, b\sin A<a<b gives two, and a\ge b gives one.

Try every case below. A calculator or a program must be in degree mode for these formulas, or the angles must be converted to radians.

How do you work with triangles in code?

Classify a triangle with integer squares, not with angles: sort the sides, check a+b>c, and compare c^2 with a^2+b^2. Keep Heron's formula exact by computing the square of the area with Fraction, and use a root only at the very end.

from fractions import Fraction
from math import sqrt, acos, degrees

def kind(a, b, c):
    a, b, c = sorted((a, b, c))
    if a + b <= c: return "not a triangle"
    sides = {1: "equilateral", 2: "isosceles", 3: "scalene"}[len({a, b, c})]
    d = a * a + b * b - c * c          # the sign of the cosine of the largest angle
    return sides + (" right" if d == 0 else " acute" if d > 0 else " obtuse")

def area_sq(a, b, c):
    s = Fraction(a + b + c, 2)
    return s * (s - a) * (s - b) * (s - c)         # Heron without the root: exact

def area_xy(p, q, r):
    return abs(p[0] * (q[1] - r[1]) + q[0] * (r[1] - p[1]) + r[0] * (p[1] - q[1])) / 2

>>> kind(3, 4, 5)
'scalene right'
>>> kind(5, 5, 8)
'isosceles obtuse'
>>> kind(2, 3, 4)
'scalene obtuse'
>>> kind(1, 2, 3)
'not a triangle'
>>> area_sq(13, 14, 15)
Fraction(7056, 1)
>>> sqrt(area_sq(13, 14, 15))
84.0
>>> area_xy((0, 0), (6, 0), (0, 8))
24.0
>>> degrees(acos(Fraction(2, 3)))                  # angle A of the 7, 8, 9 triangle
48.18968510422141
>>> a, b, c = 1e8, 1e8 - 1e-6, 1e-6               # a needle-thin triangle
>>> s = (a + b + c) / 2
>>> sqrt(max(0, s * (s - a) * (s - b) * (s - c)))  # Heron with floats
0.0
>>> sqrt(area_sq(Fraction(a), Fraction(b), Fraction(c)))   # exact on the same numbers
2.846826518921699

const [a, b, c] = [3, 4, 5]
Math.hypot(a, b)                                          // 5
Math.acos((b * b + c * c - a * a) / (2 * b * c)) * 180 / Math.PI   // angle A: 36.86989764584402
Math.sin(30 * Math.PI / 180)                              // 0.49999999999999994, not 0.5
a * a + b * b === c * c                                   // true: exact for integers, the right-angle test

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

const char* kind(int a, int b, int c) {
  int t;                                               // sort so that a <= b <= c
  if (a > b) { t = a; a = b; b = t; }
  if (b > c) { t = b; b = c; c = t; }
  if (a > b) { t = a; a = b; b = t; }
  if (a + b <= c) return "not a triangle";
  int d = a * a + b * b - c * c;                       // the sign of the cosine of the largest angle
  if (a == b && b == c) return "equilateral acute";
  if (a == b || b == c) {
    if (d == 0) return "isosceles right";
    if (d > 0) return "isosceles acute";
    return "isosceles obtuse";
  }
  if (d == 0) return "scalene right";
  if (d > 0) return "scalene acute";
  return "scalene obtuse";
}

int areaSq16(int a, int b, int c) {                    // 16 A^2 by Heron's formula, in whole numbers
  return (a + b + c) * (-a + b + c) * (a - b + c) * (a + b - c);
}

int main() {
  cout << kind(3, 4, 5) << endl;                       // scalene right
  cout << kind(5, 5, 8) << endl;                       // isosceles obtuse
  cout << kind(2, 3, 4) << endl;                       // scalene obtuse
  cout << kind(1, 2, 3) << endl;                       // not a triangle
  cout << areaSq16(13, 14, 15) / 16 << " " << sqrt(areaSq16(13, 14, 15) / 16.0) << endl;   // A^2 = 7056, A = 84
  printf("%.4f\n", acos(2.0 / 3) * 180 / acos(-1.0));  // angle A of the 7, 8, 9 triangle
  // a needle-thin triangle: the textbook form of Heron's formula loses everything, the stable form does not
  double a = 1e8, b = 1e8 - 1e-6, c = 1e-6;
  double s = (a + b + c) / 2, naive = s * (s - a) * (s - b) * (s - c);
  printf("%d\n", naive <= 0);                          // 1: nothing is left of the area under the root
  printf("%.6f\n", 0.25 * sqrt((a + (b + c)) * (c - (a - b)) * (c + (a - b)) * (a + (b - c))));   // the stable form
  return 0;
}

/* output:
scalene right
isosceles obtuse
scalene obtuse
not a triangle
7056 84
48.1897
1
2.846827
*/

The pitfalls, in order of how often they bite:

Where does the study of triangles come from?

Triangles are the oldest tool of surveying and of proof, and most of what this article covers was set down in Euclid's Elements or soon after.

• c. 600 BCE. Tradition says that Thales of Miletus measured the height of a pyramid from the length of its shadow, an early use of similar triangles.
• c. 300 BCE. Euclid's Elements, Book I, proves that the angles of a triangle add up to two right angles (I.32), and the triangle inequality (I.20); Book II gives the law of cosines in geometric form.
• c. 60 CE. Heron of Alexandria's Metrica proves the formula for the area from the three sides, which Arabic sources credit to Archimedes.
• 1765. Leonhard Euler shows that the circumcenter, centroid and orthocenter of a triangle are collinear: the Euler line.
• 1822. Karl Feuerbach describes the nine-point circle, which passes through the midpoints of the sides, the feet of the altitudes and the midpoints between the orthocenter and the vertices, and which touches the incircle.

What are the common mistakes with triangles?

The most common mistakes with triangles are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: triangles at a glance

• Angles: \angle A+\angle B+\angle C=180^\circ; an exterior angle is the sum of the two remote interior angles.
• Existence: each side is shorter than the sum of the other two; the third side is between |a-b| and a+b.
• Types: equilateral, isosceles, scalene by sides; acute, right, obtuse by c^2 against a^2+b^2 for the longest side c.
• Area: \frac12bh, Heron √(s(s-a)(s-b)(s-c)), \frac12ab\sin C, and the coordinate formula.
• Centers: centroid G (medians, 2:1), circumcenter O, orthocenter H (O,G,H on the Euler line), incenter I (r=\frac Ks).
• Congruent: SSS, SAS, ASA, AAS, RHS; similar: AA, with areas in the ratio k^2.
• Solve: law of cosines c^2=a^2+b^2-2ab\cos C, law of sines; SSA can give 0, 1 or 2 triangles.
• Code: integer squares for the right-angle test, Fraction for exact Heron, radians for sin and cos.

Glossary: key terms

Triangle

A polygon with three sides, three vertices and three interior angles that add up to 180 degrees.

Hypotenuse

The longest side of a right triangle, opposite the right angle.

Altitude

The perpendicular segment from a vertex to the line holding the opposite side; its length is the height for that base.

Median

The segment from a vertex to the midpoint of the opposite side.

Centroid

The point where the three medians meet, the balance point of the triangle, two thirds of the way along each median from its vertex.

Circumcenter

The point where the perpendicular bisectors of the sides meet, equally far from the three vertices.

Orthocenter

The point where the three altitudes of a triangle meet.

Incenter

The point where the three angle bisectors meet, equally far from the three sides and the center of the inscribed circle.

Triangle inequality

The rule that each side of a triangle is shorter than the sum of the other two sides.

Congruent triangles

Triangles with the same shape and size, so that all corresponding sides and angles are equal.

Similar triangles

Triangles with equal angles and proportional sides, the same shape but possibly different sizes.

Heron's formula

The formula that gives the area of a triangle from its three sides as the square root of s times s minus a times s minus b times s minus c.

Law of cosines

The rule that c squared equals a squared plus b squared minus twice a times b times the cosine of the angle C between them.

Law of sines

The rule that each side of a triangle divided by the sine of its opposite angle gives the same number.

Frequently asked questions

References

1. Thomas L. Heath (translator) (1908). The Thirteen Books of Euclid's Elements (2nd ed.), Book I, propositions 20, 32, 47 and 48, and Book II, propositions 12 and 13. Cambridge University Press.
2. Otto Neugebauer and Abraham Sachs (1945). Mathematical Cuneiform Texts (the tablet Plimpton 322). American Oriental Society.
3. H. S. M. Coxeter and Samuel L. Greitzer (1967). Geometry Revisited, chapters 1 and 5 (the Euler line and the nine-point circle). Mathematical Association of America.
4. William Kahan (2014). Miscalculating Area and Angles of a Needle-like Triangle. University of California, Berkeley, lecture notes.
5. Python Software Foundation. The Python Standard Library: math, mathematical functions. docs.python.org. https://docs.python.org/3/library/math.html.

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