Leg of a Right Triangle

Leg of a Right Triangle
Arm of a Right Triangle

Either of the sides in a right triangle opposite an acute angle. The legs are the two shorter sides of the triangle.

Right triangle with two shorter sides labeled "leg" and the longest side labeled "hypotenuse

See also

Key Formula

a^2 + b^2 = c^2

Where:

$a$ = One leg of the right triangle
$b$ = The other leg of the right triangle
$c$ = The hypotenuse (the longest side, opposite the right angle)

Worked Example

Problem: A right triangle has one leg of length 6 and a hypotenuse of length 10. Find the length of the other leg.

Step 1: Write the Pythagorean theorem with the known values. Let a = 6 and c = 10.

6^2 + b^2 = 10^2

Step 2: Evaluate the squares.

36 + b^2 = 100

Step 3: Subtract 36 from both sides to isolate b².

b^2 = 100 - 36 = 64

Step 4: Take the positive square root of both sides.

b = \sqrt{64} = 8

Answer: The missing leg has a length of 8. This is a 6-8-10 right triangle (a scaled version of the 3-4-5 triple).

Another Example

This example uses an isosceles right triangle (45-45-90), where both legs are equal. It shows that knowing both legs lets you find the hypotenuse directly, rather than solving for a missing leg.

Problem: Both legs of a right triangle measure 5 cm each. Find the length of the hypotenuse.

Step 1: Write the Pythagorean theorem with both legs equal to 5.

5^2 + 5^2 = c^2

Step 2: Evaluate the squares and add.

25 + 25 = c^2 \implies c^2 = 50

Step 3: Take the positive square root.

c = \sqrt{50} = 5\sqrt{2} \approx 7.07 \text{ cm}

Answer: The hypotenuse is 5√2 ≈ 7.07 cm.

Frequently Asked Questions

What is the difference between a leg and a hypotenuse?

A right triangle has three sides: two legs and one hypotenuse. The two legs are the sides that form the 90° angle, and they are always shorter than the hypotenuse. The hypotenuse is the side directly opposite the right angle and is always the longest side of the triangle.

How do you find the leg of a right triangle?

Rearrange the Pythagorean theorem. If you know the hypotenuse c and one leg a, solve for the other leg: b = √(c² − a²). If you know an angle and one side, you can also use trigonometric ratios (sine, cosine, or tangent) to find a missing leg.

Can the two legs of a right triangle be equal?

Yes. When both legs are equal, the triangle is called an isosceles right triangle. Its two acute angles are each 45°, and the hypotenuse equals the leg length times √2. This is the well-known 45-45-90 triangle.

Leg vs. Hypotenuse

	Leg	Hypotenuse
Definition	A side that forms the right angle	The side opposite the right angle
How many per triangle	2 legs	1 hypotenuse
Relative length	Always shorter than the hypotenuse	Always the longest side
Opposite angle	Opposite an acute angle (< 90°)	Opposite the right angle (= 90°)
Pythagorean theorem role	a and b in a² + b² = c²	c in a² + b² = c²

Why It Matters

Identifying the legs correctly is essential every time you use the Pythagorean theorem—mixing up a leg with the hypotenuse leads to wrong answers. Legs also appear in trigonometric ratios: for a given acute angle, one leg is the "opposite" side and the other is the "adjacent" side, which defines sine, cosine, and tangent. You will encounter legs of right triangles throughout geometry, physics (vector components), and real-world problems like finding distances and heights.

Common Mistakes

Mistake: Substituting a leg's length into c (the hypotenuse position) in the Pythagorean theorem.

Correction: Always identify c as the longest side, opposite the right angle. The two legs go in the a and b positions. If you place a leg where c belongs, you will get an incorrect or even impossible result.

Mistake: Assuming the leg labeled 'opposite' or 'adjacent' is fixed, regardless of which angle you are referencing.

Correction: The labels 'opposite' and 'adjacent' depend on which acute angle you are looking at. A leg that is opposite one acute angle is adjacent to the other. Always clarify which angle you are working with before setting up a trig ratio.

Related Terms

Hypotenuse — The third side of a right triangle, opposite the right angle
Right Triangle — The triangle that contains exactly one 90° angle
Acute Angle — Each leg is opposite one of the two acute angles
Side of a Polygon — General term for any segment forming a polygon
Pythagorean Theorem — Key formula relating both legs to the hypotenuse
Trigonometric Ratios — Sine, cosine, tangent defined using legs and hypotenuse
Isosceles Triangle — A right triangle with two equal legs is isosceles