Right Triangles: Legs and Hypotenuse
A right triangle is a triangle with one 90° angle; the side opposite the right angle is the hypotenuse, and the other two sides are the legs. In brief: the hypotenuse is always the longest side. In a right triangle with legs 3 and 4, the hypotenuse is 5, since 32 + 42 = 52.
Right triangles are the foundation of the Pythagorean theorem and of trigonometry, and they appear in every situation with perpendicular directions: a ladder against a wall, a ramp, a diagonal of a rectangle. This guide explains the names of the sides, the properties of right triangles, the common special triangles and how to find area and angles.
What is a right triangle?
A right triangle has exactly one right angle, marked with a small square at the vertex. Because the angles of any triangle add up to 180°, the other two angles must add up to 90°, so they are both acute. This means that the two acute angles are complementary: if one is 35°, the other is 55°.
A triangle cannot have more than one right angle, since two right angles already use 180° and leave nothing for the third angle.
Which side is the hypotenuse and which are the legs?
The hypotenuse is the side opposite the right angle. It is the longest side of the triangle. The two sides that form the right angle are the legs (also called catheti). The name hypotenuse comes from Greek words meaning “stretching under”, describing the side that stretches under the right angle.
When an acute angle is chosen, the legs get relative names. The leg across from the angle is the opposite side and the leg touching the angle (not the hypotenuse) is the adjacent side. These names are used in trigonometry, for instance sin A = opposite/hypotenuse.
What are the main properties of right triangles?
- The hypotenuse is longer than either leg. Since c2 = a2 + b2, we get c > a and c > b.
- The area is half the product of the legs: A = (1/2)ab, because one leg serves as the base and the other as the height.
- The perimeter is a + b + c.
- The midpoint of the hypotenuse is equally far from all three vertices. It is the center of the circumscribed circle, so the circumradius is c/2.
- The relationship between the sides is given by the Pythagorean theorem.
Worked example: area and perimeter
A right triangle has legs 6 cm and 8 cm. Its area is (1/2) · 6 · 8 = 24 cm2. The hypotenuse is √(36 + 64) = √100 = 10 cm, and the perimeter is 6 + 8 + 10 = 24 cm. The numerical value of the area and perimeter are equal here, but they have different units.
Worked example: missing angle
One acute angle is 28°. The other is 90° − 28° = 62°. The larger angle lies opposite the longer leg.
What are the special right triangles?
Isosceles right triangle (45°-45°-90°): the legs are equal, with length a, and the hypotenuse is a√2. For a = 5 the hypotenuse is 5√2 ≈ 7.07.
30°-60°-90° triangle: half of an equilateral triangle. The sides are in the ratio 1 : √3 : 2, with the shortest leg opposite 30°. For a shortest leg 4 the other leg is 4√3 ≈ 6.93 and the hypotenuse is 8.
Pythagorean triples: right triangles with whole-number sides, such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Multiples of a triple, such as 6-8-10, are right triangles as well.
Right triangle summary table
| Triangle | Legs | Hypotenuse | Area | Acute angles |
|---|---|---|---|---|
| 3-4-5 | 3 and 4 | 5 | 6 | about 36.9° and 53.1° |
| 5-12-13 | 5 and 12 | 13 | 30 | about 22.6° and 67.4° |
| 8-15-17 | 8 and 15 | 17 | 60 | about 28.1° and 61.9° |
| 7-24-25 | 7 and 24 | 25 | 84 | about 16.3° and 73.7° |
| 1-1-√2 | 1 and 1 | √2 | 1/2 | 45° and 45° |
| 1-√3-2 | 1 and √3 | 2 | √3/2 | 30° and 60° |
Where does the idea come from?
Right angles and the 3-4-5 triangle were used by builders in ancient civilizations. Babylonian clay tablets show that scribes knew many sets of whole-number sides, and the Egyptians are often said to have used knotted ropes with 3-4-5 spacing for right angles, though direct evidence for this is limited. Euclid’s Elements gives the first systematic treatment of right triangles in Greek geometry.
Where are right triangles used today?
Construction relies on right triangles to check squareness: if the sides of a corner measure 3 and 4 units and the diagonal measures 5, the corner is a right angle. Ramps and roof pitches are right triangles. Surveyors and navigators use them to find heights and distances. Screen sizes are given as diagonals, which are hypotenuses.
Which mistakes are common?
Calling a leg the hypotenuse
The hypotenuse is always opposite the right angle, regardless of how the triangle is turned. A rotated triangle keeps the same hypotenuse.
Using the hypotenuse as the base for the area
The formula (1/2)ab uses the two legs. Using the hypotenuse requires the corresponding height.
Assuming the 3-4-5 pattern for any triangle
Only a triangle whose sides satisfy a2 + b2 = c2 is a right triangle. For example 4, 5 and 6 is not.
Forgetting units
Lengths and perimeters use units such as cm, while areas use cm2.