Right Triangle

Right Triangle

A triangle which has a right (90°) interior angle.

Right triangle with a 90° angle at bottom-left, two sides labeled "leg," and the longest side labeled "hypotenuse.

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 side opposite the 90° angle)

Worked Example

Problem: A right triangle has legs of length 6 and 8. Find the length of the hypotenuse.

Step 1: Identify the known values. The two legs are a = 6 and b = 8. The hypotenuse c is unknown.

a = 6, \quad b = 8

Step 2: Write the Pythagorean Theorem and substitute the known values.

6^2 + 8^2 = c^2

Step 3: Compute the squares and add them together.

36 + 64 = c^2 \implies 100 = c^2

Step 4: Take the positive square root of both sides to solve for c.

c = \sqrt{100} = 10

Answer: The hypotenuse is 10 units long.

Another Example

This example differs by working backward from the hypotenuse to find a missing leg, and then uses trigonometry (SOHCAHTOA) to find the acute angles — combining both key right-triangle tools.

Problem: A right triangle has a hypotenuse of length 13 and one leg of length 5. Find the missing leg and the two acute angles (to the nearest degree).

Step 1: Use the Pythagorean Theorem to find the missing leg. Let a = 5 and c = 13.

5^2 + b^2 = 13^2 \implies 25 + b^2 = 169

Step 2: Solve for b.

b^2 = 169 - 25 = 144 \implies b = 12

Step 3: Find the angle opposite the leg of length 5 using the sine ratio (SOHCAHTOA).

\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13} \implies \theta = \sin^{-1}\!\left(\frac{5}{13}\right) \approx 23°

Step 4: Since the three angles of any triangle sum to 180°, the remaining acute angle is:

180° - 90° - 23° = 67°

Answer: The missing leg is 12 units long, and the two acute angles are approximately 23° and 67°.

Frequently Asked Questions

How do you know which side is the hypotenuse in a right triangle?

The hypotenuse is always the side directly opposite the 90° angle. It is also always the longest of the three sides. When using the Pythagorean Theorem, the hypotenuse is the side that stands alone on one side of the equation (c in a² + b² = c²).

Can a right triangle also be isosceles or scalene?

Yes. A right triangle is isosceles when its two legs are equal in length, producing angles of 45°, 45°, and 90°. Most right triangles, however, are scalene, meaning all three sides have different lengths. A right triangle can never be equilateral because an equilateral triangle has three 60° angles.

What are the most common right triangle side ratios?

The most frequently used Pythagorean triples are 3-4-5, 5-12-13, and 8-15-17. Any multiple of these also works (for example, 6-8-10 is a multiple of 3-4-5). Two special right triangles defined by their angles are the 45-45-90 triangle (sides in ratio 1 : 1 : √2) and the 30-60-90 triangle (sides in ratio 1 : √3 : 2).

Right Triangle vs. Oblique Triangle

	Right Triangle	Oblique Triangle
Definition	Has exactly one 90° angle	Has no 90° angle (all angles are acute, or one is obtuse)
Key formula	Pythagorean Theorem: a² + b² = c²	Law of Cosines: c² = a² + b² − 2ab cos C
Trigonometry approach	Basic SOHCAHTOA ratios apply directly	Requires Law of Sines or Law of Cosines
Angle sum of acute angles	The two non-right angles always add to 90°	All three angles add to 180°, but no single angle is 90°

Why It Matters

Right triangles are the foundation of trigonometry — sine, cosine, and tangent are all defined using the sides of a right triangle. You will encounter them constantly in geometry, physics (resolving forces into components), architecture, and navigation. Standardized tests like the SAT and ACT frequently test your ability to apply the Pythagorean Theorem and special right-triangle ratios.

Common Mistakes

Mistake: Using a leg instead of the hypotenuse as c in the Pythagorean Theorem.

Correction: Always assign c to the side opposite the 90° angle (the longest side). The two legs go in the a² + b² positions. Mixing this up gives a wrong answer every time.

Mistake: Assuming any triangle with a large angle is a right triangle.

Correction: A triangle is a right triangle only if one angle is exactly 90°. An 89° or 91° angle does not qualify. To verify, check whether a² + b² equals c² for the three side lengths.

Related Terms

Triangle — General category that includes right triangles
Right Angle — The 90° angle that defines a right triangle
Interior Angle — Angles inside a triangle, one of which is 90°
Pythagorean Theorem — Key formula relating the sides of a right triangle
Trigonometry — Branch of math built on right-triangle ratios
SOHCAHTOA — Mnemonic for sine, cosine, and tangent ratios
Hypotenuse — The longest side, opposite the right angle