Right Triangle: Parts, Legs and Hypotenuse

A right triangle (also called a right-angled triangle) is a triangle with one angle of exactly 90°. The two sides that form the right angle are the legs, and the side opposite the right angle is the hypotenuse, which is always the longest side. The other two angles are acute and add up to 90°.

Being able to recognize these parts quickly, in any position, is the skill everything else depends on. The Pythagorean theorem, trigonometry and the distance formula all start with the same question: which side is the hypotenuse?

Parts of a right triangle

Right angle

An angle of exactly 90°, a "square corner" like the corner of a sheet of paper. Diagrams mark it with a small square instead of an arc. A triangle can have at most one.

Legs

The two sides that meet at the right angle, usually called a and b. They are perpendicular to each other, so either one can serve as the base and the other as the height.

Hypotenuse

The side opposite the right angle, usually called c. It is the only side that does not touch the right angle, and it is always the longest. See the hypotenuse page for how to calculate it.

Acute angles

The two other angles, each less than 90°. Together they make exactly 90°.

Vertices

The three corners, named with capital letters. By convention the right angle is at C, and each side takes the lowercase letter of the corner across from it.

Opposite and adjacent

Words used in trigonometry. Seen from angle A, leg a is the opposite side and leg b is the adjacent side; from angle B the roles swap. The hypotenuse is the hypotenuse from both.

How to recognize a right triangle

A triangle that looks right-angled may be 88° or 93°. You need evidence. Any one of these is enough:

• The right-angle mark. A small square in a corner means exactly 90°.
• A stated angle. The problem says one angle is 90°, or gives two angles that add up to 90°. If two angles are 35° and 55°, the third is 180° − 35° − 55° = 90°.
• Perpendicular lines. A wall meeting level ground, the corner of a rectangle, grid lines on graph paper or the axes of a coordinate plane.
• The side lengths. If the squares of the two shorter sides add up to the square of the longest, the triangle is right-angled. Sides 6, 8 and 10 pass, because 36 + 64 = 100; sides 6, 8 and 11 do not, because 100 ≠ 121. This test is the converse of the Pythagorean theorem, and the right triangle checker runs it for you.
• Slopes on a grid. Two sides are perpendicular when their slopes multiply to −1, for example slopes 2 and −1 over 2.

Finding the hypotenuse in any orientation

Textbooks love to turn, flip and stretch right triangles so that the hypotenuse is not the slanted side at the top right. The rule never changes: find the square corner, then look straight across it. The side you land on is the hypotenuse.

Three quick checks help when a diagram is cluttered:

• The hypotenuse never touches the right-angle square. Both legs do.
• The hypotenuse is the longest side. If the side you picked is shorter than another side, you picked a leg.
• Flipping a triangle over, like a page, does not change which side is which. A mirror image of a right triangle is still a right triangle with the same hypotenuse.

Why the hypotenuse is always the longest side

It is worth knowing why this is true, not just that it is. Picture swinging each leg around a corner until it lies along the hypotenuse:

The picture suggests two facts. Here are short proofs of each.

Together the two claims trap the hypotenuse: it is longer than the longer leg but shorter than the two legs combined. That makes a quick reasonableness check for any answer, used in how to do the Pythagorean theorem.

The two acute angles add up to 90°

Because the three angles of a triangle total 180° and the right angle takes 90°, the other two angles always add up to exactly 90°. Angles with this property are called complementary.

So if you know one acute angle, you know the other: an angle of 35° comes with one of 55°, and an angle of 62.5° comes with one of 27.5°. In the 3-4-5 triangle the angles are about 36.87° and 53.13°, which add up to 90°. This also explains two facts that surprise some students: a right triangle cannot have an obtuse angle, and it cannot have a second right angle, because there is no room left in the 180°. Finding the angles from the sides uses trigonometry; see when to use the Pythagorean theorem or trig.

Area and perimeter of a right triangle

The legs make right triangles the easiest triangles to measure. Two copies of a right triangle fit together into a rectangle with sides a and b (as in the first shape in the figure further down), so the triangle has half its area:

The perimeter is the sum of the three sides, a + b + c. If you only know the legs, you need the theorem to find c first.

You can also measure area using the hypotenuse as the base. The height to the hypotenuse, h, then satisfies 1 over 2ch = 1 over 2ab, so h = ab over c. For the 5-12-13 triangle, h = 60/13 ≈ 4.62 cm. That height splits the triangle into two smaller right triangles, the subject of the geometric mean theorem.

Right triangles hiding in other shapes

Many problems do not show a right triangle at all; you have to find one. These are the most common hiding places:

• Rectangles. A diagonal cuts a rectangle into two identical right triangles; the diagonal is their hypotenuse. This is how screen sizes and the diagonal of a rectangle are calculated.
• Squares. A diagonal makes two isosceles right triangles with angles 45°, 45° and 90°.
• Isosceles and equilateral triangles. The height from the top corner meets the base at 90° and cuts it in half, giving two right triangles. That is how you find the height of a triangle.
• Rhombuses and kites. Their diagonals cross at right angles, cutting the shape into right triangles.
• Semicircles. Any triangle drawn with a diameter as one side and its third corner on the circle has a right angle at that corner. This is Thales' theorem.
• Coordinate grids. Moving across and then up between two points traces two legs; the straight line between the points is the hypotenuse. That is the distance formula.

Special right triangles

Three families of right triangles come up so often that it pays to know them by heart:

• The 45-45-90 triangle: half a square, with sides in the ratio 1 : 1 : √2.
• The 30-60-90 triangle: half an equilateral triangle, with sides in the ratio 1 : √3 : 2.
• Pythagorean triples such as 3-4-5, 5-12-13 and 8-15-17: right triangles whose three sides are all whole numbers.

To solve any right triangle from two pieces of information, sides or angles, use the right triangle calculator.