What are the legs and the hypotenuse of a right triangle?

Find the right angle first

The right angle is marked with a small square at the corner. It is the first thing to look for in any diagram, because everything else follows from it: the side facing it is the hypotenuse, and the two sides meeting at it are the legs.

The legs are perpendicular to one another — that perpendicularity is the right angle itself. Nothing about which side looks longer or sits lower on the page enters into it.

Why the hypotenuse is always longest

In any triangle, the longer side lies opposite the larger angle. The right angle is ninety degrees, and the other two must share the remaining ninety — so each of them is smaller than ninety.

That makes the right angle the largest in the triangle, and the side facing it the longest. This is not a fact to memorise but a check to use: if you calculate a hypotenuse that comes out shorter than a leg, the working is wrong and there is no point continuing.

The drawing is rarely upright

In exams the triangle is almost never drawn with its legs horizontal and vertical. It is rotated, flipped, or buried inside a larger shape, and the eye automatically looks for the bottom side or the longest one on the page.

The rule does not depend on orientation. Find the small square, mark the side facing it as the hypotenuse, and only then write the equation. Marking the diagram in pencil beats any attempt to hold it in your head.

When the triangle is hidden inside another shape — a diagonal of a rectangle, the height of an isosceles triangle — redraw it on its own at the side. A triangle standing alone on the page almost removes the possibility of this mistake.

Worked examples

1. A right triangle has sides 6, 8 and 10. Which is the hypotenuse?

   1. The hypotenuse is always the longest side
   2. Compare the three: 10 is greater than 8 and than 6
   3. So 10 is the hypotenuse, and 6 and 8 are the legs

   Answer: The hypotenuse is 10; the legs are 6 and 8

2. Could a right triangle have sides 5 and 12 with a hypotenuse of 11?

   1. The hypotenuse must be the longest side
   2. Here 12 is greater than 11
   3. So a side would be longer than the hypotenuse, which is impossible

   Answer: No. If 11 is the hypotenuse it cannot be shorter than 12

3. A rectangle is 8 long and 6 wide, with a diagonal drawn. What triangle appears, and which side is the hypotenuse?

   1. The diagonal cuts the rectangle into two triangles
   2. Every corner of a rectangle is a right angle, so each triangle is right-angled
   3. The legs are the length and the width, and the diagonal faces the right angle

   Answer: A right triangle with legs 8 and 6, and the diagonal as hypotenuse

Common mistakes

Assuming the slanted side in the drawing is the hypotenuse

When the triangle is rotated, the side that looks slanted can be a leg. The right angle decides, not the direction on the page, so look for the small square.

Putting the hypotenuse where a leg belongs in the equation

The hypotenuse stands alone on one side of the theorem and the two legs on the other. Swapping them gives an answer that looks plausible and is not, which is why checking that the hypotenuse came out longest matters so much.

Identifying the sides by the order they appear in the question

The question is under no obligation to list them in any order. The diagram decides, and a triangle pulled out of a larger figure is worth redrawing before you write anything down.

More in The Pythagorean Theorem

• What is a square number and what is a square root?
• What does Pythagoras' theorem say and why is it true?
• How do you find a leg when the hypotenuse is known?
• How do you check whether a triangle is right-angled?
• How do you find the distance between two points and a diagonal in a rectangle?