Right Triangle Formulas (Pythagorean Theorem, Area)

A right triangle has one 90° angle. The Pythagorean theorem relates the legs a and b to the hypotenuse c: c = √(a² + b²). Its area is A = ½·a·b, half the product of the two legs.

Right Triangle formulas

Hypotenuse: $c = \sqrt{a^2 + b^2}$
Area: $A = \tfrac{1}{2} a b$
Perimeter: $P = a + b + c$

where a, b = the two legs, c = hypotenuse (longest side), A = area

Hypotenuse

c = \sqrt{a^2 + b^2}

Worked example

A right triangle with legs 3 and 4 has hypotenuse 5 — the classic 3-4-5 triangle.

c = \sqrt{a^2 + b^2} = \sqrt{9 + 16} = \sqrt{25} = 5

Area

A = \tfrac{1}{2} a b

Perimeter

P = a + b + c

Interactive right triangle explorer

Drag a slider (or type a value, or use the arrow keys) and watch the diagram, the results, and the full step-by-step substitution update live.

Side a a

Base / side b b

Hypotenuse c 5units

Area A 6units²

Perimeter P 12units

Step-by-step substitution

c = √(a² + b²) = √(9 + 16) = 5
A = ½ · a · b = ½ · 3 · 4 = 6
P = a + b + c = 3 + 4 + 5 = 12

Classroom use: right triangle formulas

Who it's for: Grades 8–11 · Students learning or applying the Pythagorean theorem, and high-school students beginning right-triangle trigonometry.

Learning objectives

After using this page, students will be able to:

Apply a² + b² = c² to find an unknown leg or the hypotenuse.
Identify Pythagorean triples and use them to check answers quickly.
Compute the area of a right triangle using its two legs as base and height.

Suggested classroom activity

Start at the 3-4-5 triangle in the explorer, then scale both legs and let students verify that 6-8-10 and 9-12-15 also work. The step-by-step substitution panel exposes each stage of the arithmetic, so you can pause on the squaring step where most errors occur.

Standards alignment

Aligned to the Common Core State Standards for Mathematics. Codes are given exactly as published so they can be matched against state crosswalks.
Code	Standard
`8.G.B.7`	Apply the Pythagorean theorem to find unknown side lengths in right triangles.
`HSG.SRT.C.8`	Use trigonometric ratios and the Pythagorean theorem to solve right triangles.
`HSG.MG.A.1`	Use geometric shapes, their measures, and their properties to describe objects.

Free to use and adapt in class under the terms on our license page — no account, no sign-up, no ads. Content last reviewed August 2026.

Related & next steps

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Frequently asked questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right triangle, a² + b² = c², where c is the hypotenuse. So c = √(a² + b²). For legs 6 and 8, the hypotenuse is 10.

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