MENUClassification of Triangles
Triangles are classified in two ways: by their sides and by their angles. The full name of a triangle uses both, for example "right isosceles triangle" or "obtuse scalene triangle".
On drawings, equal sides are marked with the same number of small ticks, and equal angles with the same number of arcs. A small square marks a right angle.
In triangle $ABC$ the marks say that ${AC=BC}$ and ${\angle A=\angle B}.$ In triangle $DEF$ the angle at $D$ is a right angle.
On this page: By sides · By angles · Possible combinations · Type from the side lengths · Examples · Common mistakes · Practice
Classification by Sides
- Scalene triangle: all three sides have different lengths. Then its three angles are different too.
- Isosceles triangle: two sides are equal. The equal sides are the legs, and the third side is the base. The angles at the base, the base angles, are equal.
- Equilateral triangle: all three sides are equal. Then all three angles are equal too, each ${60^\circ}.$
An equilateral triangle is a special isosceles triangle: it has two equal sides, and its third side is equal to them too. So everything that is true for every isosceles triangle is also true for an equilateral one.
Classification by Angles
- Acute triangle: all three angles are acute (smaller than ${90^\circ}$).
- Right triangle: one angle is a right angle (${90^\circ}$). The side opposite it is the hypotenuse, and the other two sides are the legs.
- Obtuse triangle: one angle is obtuse (greater than ${90^\circ}$).
Every triangle has at least two acute angles, because its three angles add up to ${180^\circ}.$ A right triangle and an obtuse triangle have two acute angles too, but that does not make them acute. Look at the largest angle. If it is acute, the triangle is acute; if it is right, the triangle is right; if it is obtuse, the triangle is obtuse.
Which Combinations Are Possible?
Let us name each triangle by its most special type by sides, so that an equilateral triangle is not counted again as isosceles. Then every triangle has one type by sides and one type by angles, and seven of the nine combinations exist. The table gives an example of the angles for each.
| Acute | Right | Obtuse | |
|---|---|---|---|
| Scalene | ${50^\circ},$ ${60^\circ},$ ${70^\circ}$ | ${30^\circ},$ ${60^\circ},$ ${90^\circ}$ | ${20^\circ},$ ${40^\circ},$ ${120^\circ}$ |
| Isosceles | ${70^\circ},$ ${70^\circ},$ ${40^\circ}$ | ${45^\circ},$ ${45^\circ},$ ${90^\circ}$ | ${30^\circ},$ ${30^\circ},$ ${120^\circ}$ |
| Equilateral | ${60^\circ},$ ${60^\circ},$ ${60^\circ}$ | — | — |
A dash means that the combination is impossible. An equilateral triangle is always acute, because all its angles are ${60^\circ}.$ In a right or obtuse isosceles triangle, the right or obtuse angle must be the vertex angle: two base angles of ${90^\circ}$ or more would already add up to at least ${180^\circ}.$
How to Find the Type from the Side Lengths
By sides it is easy: count the equal sides. By angles, call the longest side $c$ (whichever side it is) and the other two sides $a$ and $b,$ and compare ${c^2}$ with ${a^2+b^2}.$
${c^2\lt a^2+b^2}$ means the triangle is acute;
${c^2=a^2+b^2}$ means the triangle is right (the converse of the Pythagorean theorem);
${c^2\gt a^2+b^2}$ means the triangle is obtuse.
Why does this work? By the law of cosines ${c^2=a^2+b^2-2ab\cos\gamma},$ where $\gamma$ is the angle opposite the longest side, that is, the largest angle. The number ${\cos\gamma}$ is positive, zero or negative exactly when $\gamma$ is acute, right or obtuse.
Before you classify, check that the triangle exists at all: the longest side must be shorter than the sum of the other two.
- Sides $5,$ $6$ and $7.$ Since ${7^2=49\lt 5^2+6^2=61},$ the triangle is acute; all sides are different, so it is an acute scalene triangle.
- Sides $6,$ $8$ and $10.$ Since ${10^2=100=6^2+8^2},$ it is a right scalene triangle.
- Sides $5,$ $5$ and $8.$ Since ${8^2=64\gt 5^2+5^2=50},$ it is an obtuse isosceles triangle.
Examples
Example 1. Classify this triangle by its sides and by its angles.
Solution. The marks show two equal sides, ${AB=BC},$ so by sides the triangle is isosceles. The square at $B$ marks a right angle, so by angles the triangle is right. It is a right isosceles triangle. Its acute angles are equal: ${\angle A=\angle C=45^\circ}.$
Example 2. In triangle $ABC$ the angle at $A$ is ${40^\circ}$ and the angle at $B$ is ${110^\circ}.$ Classify the triangle.
Solution. The third angle is ${180^\circ-40^\circ-110^\circ=30^\circ}.$ The three angles are different, so the three sides are different too: the triangle is scalene. The angle at $B$ is greater than ${90^\circ},$ so the triangle is obtuse. It is an obtuse scalene triangle.
Example 3. Two angles of a triangle are ${35^\circ}$ and ${55^\circ}.$ Classify the triangle.
Solution. The third angle is ${180^\circ-35^\circ-55^\circ=90^\circ}.$ All three angles are different, so it is a right scalene triangle.
Example 4. The angles of a triangle are in the ratio ${2:3:4}.$ Classify the triangle.
Solution. ${2k+3k+4k=180^\circ},$ so ${k=20^\circ},$ and the angles are ${40^\circ},$ ${60^\circ}$ and ${80^\circ}.$ They are all acute and all different: an acute scalene triangle.
Example 5. Classify the triangle with sides 8 cm, 8 cm and 13 cm.
Solution. The triangle exists, because ${13\lt 8+8}.$ Two sides are equal, so it is isosceles. The longest side is 13, and ${13^2}={169}\gt{8^2+8^2}={128},$ so the angle opposite it is obtuse. It is an obtuse isosceles triangle.
Common Mistakes
| Mistake | Wrong | Right |
|---|---|---|
| Classifying by an acute angle | The angles are ${30^\circ},$ ${60^\circ},$ ${90^\circ}.$ There are acute angles, so the triangle is acute | The largest angle is ${90^\circ},$ so the triangle is right |
| Thinking that a right triangle cannot be isosceles | A right triangle always has three different sides | The angles ${45^\circ},$ ${45^\circ},$ ${90^\circ}$ give a right isosceles triangle |
| Squares compared with the wrong side | Sides 5, 12, 13. Since ${5^2\lt 12^2+13^2},$ the triangle is acute | Use the longest side: ${13^2=5^2+12^2},$ so the triangle is right |
| No check that the triangle exists | Sides 2, 3, 6. Since ${6^2\gt 2^2+3^2},$ the triangle is obtuse | ${6\gt 2+3},$ so there is no such triangle |
| An obtuse base angle | An isosceles triangle with base angles of ${100^\circ}$ | Impossible: an obtuse angle of an isosceles triangle can only be the vertex angle |
Practice Problems
Classify each triangle by its sides and by its angles, then open the answer.
1. The angles are ${30^\circ},$ ${60^\circ}$ and ${90^\circ}.$
A right scalene triangle.
2. Two angles are ${50^\circ}$ and ${65^\circ}.$
The third angle is ${65^\circ}.$ Two angles are equal, so two sides are equal: an acute isosceles triangle.
3. The sides are 9, 12 and 15.
${15^2=225=9^2+12^2},$ so it is a right scalene triangle.
4. The sides are 5, 5 and 8.
${8^2=64\gt 5^2+5^2=50},$ so it is an obtuse isosceles triangle.
5. The sides are 6, 7 and 8.
${8^2=64\lt 6^2+7^2=85},$ so it is an acute scalene triangle.
6. The sides are 7, 7 and 7.
An equilateral triangle; its angles are ${60^\circ},$ so it is acute.
7. The sides are 2, 3 and 6.
There is no such triangle, because ${6\gt 2+3}.$
8. Can a right triangle be equilateral?
No. All angles of an equilateral triangle are ${60^\circ},$ so none of them is a right angle.
9. One angle of an isosceles triangle is ${100^\circ}.$ Find the other angles and classify the triangle.
${100^\circ}$ must be the vertex angle, so the base angles are ${40^\circ}$ each. It is an obtuse isosceles triangle.
10. The angles are in the ratio ${1:1:2}.$
${k+k+2k=180^\circ},$ so the angles are ${45^\circ},$ ${45^\circ}$ and ${90^\circ}:$ a right isosceles triangle.
Test yourself: Types of Triangles Based on Angles and Sides.
More about triangles: Triangles · Altitude · Median · Congruence
Types of Triangles