Hypotenuse

Hypotenuse

The side of a right triangle opposite the right angle. Note: The hypotenuse is the longest side of a right triangle.

Right triangle with two legs (vertical and horizontal) and hypotenuse labeled along the longest side opposite the right angle.

See also

Leg of a right triangle

Key Formula

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

Where:

$c$ = The length of the hypotenuse (the side opposite the right angle)
$a$ = The length of one leg of the right triangle
$b$ = The length of the other leg of the right triangle

Worked Example

Problem: A right triangle has legs of length 3 and 4. Find the length of the hypotenuse.

Step 1: Write the Pythagorean theorem.

c^2 = a^2 + b^2

Step 2: Substitute the leg lengths into the formula.

c^2 = 3^2 + 4^2

Step 3: Square each leg and add the results.

c^2 = 9 + 16 = 25

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

c = \sqrt{25} = 5

Answer: The hypotenuse is 5 units long.

Another Example

This example works in reverse — you already know the hypotenuse and need to find a missing leg. This is a common variation students encounter on tests.

Problem: The hypotenuse of a right triangle is 13 and one leg is 5. Find the length of the other leg.

Step 1: Start with the Pythagorean theorem and solve for the unknown leg.

a^2 + b^2 = c^2

Step 2: Substitute the known values. Let the unknown leg be b.

5^2 + b^2 = 13^2

Step 3: Simplify the squares.

25 + b^2 = 169

Step 4: Isolate b² by subtracting 25 from both sides.

b^2 = 169 - 25 = 144

Step 5: Take the square root to find b.

b = \sqrt{144} = 12

Answer: The other leg is 12 units long.

Frequently Asked Questions

How do you find the hypotenuse of a right triangle?

Use the Pythagorean theorem: square each leg, add the two results together, and then take the square root. The formula is

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

. For example, if the legs are 6 and 8, the hypotenuse is

\sqrt{36 + 64} = \sqrt{100} = 10

Why is the hypotenuse always the longest side?

In any triangle, the longest side is always opposite the largest angle. A right angle measures 90°, and since the three angles of a triangle must add up to 180°, neither of the other two angles can be 90° or more. This makes the right angle the largest angle, so the side opposite it — the hypotenuse — must be the longest side.

Can the hypotenuse be a leg of a right triangle?

No. The hypotenuse and the legs are distinct parts of a right triangle. The two legs are the sides that form the right angle, while the hypotenuse is the side across from it. Every right triangle has exactly one hypotenuse and exactly two legs.

Hypotenuse vs. Leg of a Right Triangle

	Hypotenuse	Leg of a Right Triangle
Definition	The side opposite the right angle	Either of the two sides that form the right angle
How many per triangle	Exactly 1	Exactly 2
Relative length	Always the longest side	Always shorter than the hypotenuse
Role in Pythagorean theorem	c in c² = a² + b²	a and b in c² = a² + b²
Position	Opposite the 90° angle	Adjacent to the 90° angle

Why It Matters

The hypotenuse appears throughout geometry, trigonometry, and physics. In trigonometry, sine and cosine are both defined as ratios involving the hypotenuse (

\sin\theta = \text{opposite}/\text{hypotenuse}

\cos\theta = \text{adjacent}/\text{hypotenuse}

). You also use it whenever you calculate a straight-line distance between two points on a coordinate plane, since the distance formula is derived directly from the Pythagorean theorem.

Common Mistakes

Mistake: Placing the hypotenuse on the wrong side of the Pythagorean theorem — for instance, writing

c^2 + a^2 = b^2

where c is the hypotenuse.

Correction: The hypotenuse must always stand alone on one side of the equation:

c^2 = a^2 + b^2

. The sum of the squares of the two legs equals the square of the hypotenuse, not the other way around.

Mistake: Assuming the hypotenuse is the vertical or bottom side of the triangle based on how the figure is drawn.

Correction: The hypotenuse is defined by its position relative to the right angle, not by its orientation on the page. Always look for the right-angle marker (the small square) first, then identify the side across from it.

Related Terms

Right Triangle — The triangle that contains a hypotenuse
Right Angle — The 90° angle opposite the hypotenuse
Leg of a Right Triangle — The other two sides forming the right angle
Pythagorean Theorem — The formula used to calculate the hypotenuse
Side of a Polygon — General term for any segment of a polygon
Distance Formula — Derived from the Pythagorean theorem using the hypotenuse
Trigonometry — Uses the hypotenuse to define sine and cosine