Triangles are classified twice over, once by the lengths of their sides and once by the size of their angles. The two systems are largely independent, which is why a triangle is usually described with a word from each, such as an isosceles right triangle.
By sides
| Name | Sides | Angles that follow |
|---|---|---|
| Equilateral | all three equal | all three are 60 degrees |
| Isosceles | exactly two equal | the two angles opposite those sides are equal |
| Scalene | no two equal | no two angles equal |
By angles
| Name | Angles |
|---|---|
| Acute | all three angles less than 90 degrees |
| Right | one angle exactly 90 degrees |
| Obtuse | one angle greater than 90 degrees |
A triangle can have at most one right angle or one obtuse angle, because two of them would use up 180 degrees or more before the third angle even appears.
The angle sum
The three angles of any triangle add to 180 degrees, whatever its shape. That single fact answers most missing angle questions.
Finding the third angle
first angle 38 second angle 71 180 - 38 - 71 = 71 degrees two angles are equal, so the triangle is isosceles
Which lengths can make a triangle
Three lengths only close into a triangle if the two shorter ones together are longer than the longest. Lengths of 3, 4 and 9 cannot, because 3 + 4 = 7 is less than 9, and the two short sides cannot reach across.
Common mistakes
Giving only one name
Most triangles deserve two words, one from each classification.
Assuming from the picture
Sides that look equal in a sketch are not equal unless marked. Angles that look right are not right unless marked with a square.
Check yourself
A triangle has angles 90, 45 and 45. Name it twice.
Right by angle, isosceles by side, since the two equal angles sit opposite two equal sides.
Can sides of 5, 5 and 11 form a triangle?
No. 5 + 5 = 10, which is less than 11, so the short sides cannot meet.
Two angles of a triangle are 33 and 47. What is the third, and what kind of triangle is it?
180 - 33 - 47 = 100 degrees, so it is obtuse and scalene.
Why is every equilateral triangle also acute?
Because all three angles are equal and add to 180, so each is 60 degrees, which is less than 90.
See also
Sources
- Triangle classification, the angle sum and the triangle inequality as taught in United States elementary geometry. Diagrams are drawn for this site.