Triangle formula chart
The key triangle formulas in one place: area, perimeter, the Pythagorean theorem, the angle sum, and the side ratios of the two special right triangles. Match your situation to the row and plug in what you know.
Triangle formulas
| Quantity | Formula |
|---|---|
| Area (base and height) | A = ½ × b × h |
| Area (Heron, three sides) | A = √( s(s−a)(s−b)(s−c) ), s = (a+b+c)/2 |
| Perimeter | P = a + b + c |
| Pythagorean theorem | a² + b² = c² (right triangle) |
| Angle sum | A + B + C = 180° |
Special right triangles
| Triangle | Side ratio (short : long : hypotenuse) |
|---|---|
| 45-45-90 | 1 : 1 : √2 |
| 30-60-90 | 1 : √3 : 2 |
Triangle types
| Type | Description |
|---|---|
| Equilateral | All three sides equal, all angles 60° |
| Isosceles | Two sides equal, two angles equal |
| Scalene | All sides and angles different |
| Right | One angle is exactly 90° |
The angles of any triangle always add to 180 degrees. Use the base-and-height area formula when you know the height, and Heron formula when you only know the three side lengths. The special right triangles let you find exact side lengths from a single side.
Need more geometry?
See the Area Formula Chart and the Distance Formula.
Finding the area
If you know the base and the perpendicular height, the area is half their product. If you only have the three side lengths, Heron formula works: compute the semi-perimeter s as half the perimeter, then take the square root of s times each of s minus a side. The two methods always agree.
Special right triangles
Two right triangles appear so often they are worth memorizing. The 45-45-90 triangle has legs in the ratio 1 to 1 with a hypotenuse of root two. The 30-60-90 triangle has sides in the ratio 1 to root three to 2. Knowing one side of either lets you write down the others instantly, which is the basis of the unit circle values.
FAQ
Do the angles of a triangle always add to 180 degrees?
Yes, for any flat (Euclidean) triangle the three interior angles sum to exactly 180 degrees.
How do I find the area from three sides?
Use Heron formula: find s = (a+b+c)/2, then area = √(s(s−a)(s−b)(s−c)).
What are the 30-60-90 side ratios?
1 : √3 : 2, for the sides opposite the 30, 60, and 90 degree angles respectively.