Special Right Triangles Formulas
Before learning the special right triangles formulas, let us recall what are special right triangles. There are two special right triangles with angles measures as 45, 45, 90 degrees and 30, 60, 90 degrees. The sides of these triangles are in particular ratios and are known as Pythagorean triplets. Let us learn the special right triangles formulas along with a few solved examples.
What Are the Special Right Triangles Formulas?
The special right triangles formulas give the ratio of the sides of a special right triangle. The base, height, and hypotenuse of a right triangle with the angles 45, 45, and 90 degrees are in a ratio of
1: 1: √2
The base, height, and hypotenuse of a right triangle with the angles 30, 60, and 90 degrees are in a ratio of
1: √3: 2
Here, the order remains the same. Let us see the applications of the special right triangles formulas in the following section.
Solved Examples Using Special Right Triangles Formulas

Example 1: Find the other two sides of the rightangled triangle if the base of the triangle is 5√3 and angles measure 30, 60, and 90 degrees.
Solution: To find: Length of the sides of the triangle.
Base = 5√3 (given)
Using special right triangles formulas,
Base, height, and hypotenuse of a triangle with the angles 30, 60, and 90 degrees are in a ratio of 1:√3: 2
Let, base = 5√3 = x
Then, height = (5√3)√3= 5 *3 = 15
And, hypotenuse = 2x = 2*5√3 = 10√3
Answer: The height of the triangle = 15, and hypotenuse of the triangle = 10√3.

Example 2: Find the other two sides of the rightangled triangle if the base of the triangle is 5 and angles measure 45, 45, and 90 degrees.
Solution: To find: Length of the sides of the triangle.
Base = 5 (given)
Using special right triangles formulas,
Base, height, and hypotenuse of a triangle with the angles 45, 45, and 90 degrees are in a ratio of √2
Let, base = 5 = x
Then, height = x = 5
And hypotenuse = √2x =√2*5= 5√2
Answer: The height of the triangle = 5, and hypotenuse of the triangle = 5√2.