How to Find the Height of a Right Triangle to the Hypotenuse

Geometry Lessons

The short answer

h = ab / c

Multiply the two legs and divide by the hypotenuse. This height is a fourth length, shorter than either leg, and it is not one of the sides.

On a right triangle the two legs are already perpendicular to each other, so each one serves as the height for the other. That fact is so convenient that people start assuming the height of a right triangle is always a side. Stand the triangle on its hypotenuse and that assumption breaks immediately.

Where the formula comes from

It is the area formula used twice. Take the two legs as base and height, and the area is half of a times b. Now take the hypotenuse as the base instead, and the area is half of c times the height you are looking for. The triangle has not changed, so those two expressions must be equal.

Set them equal and the halves cancel, leaving ab = ch. Divide by c and you have h = ab / c. Nothing new was needed. The whole derivation is just refusing to let the area depend on which side you chose to stand the triangle on.

1

Multiply the two legs

These are the sides meeting at the right angle. Their product is twice the area.

2

Divide by the hypotenuse

The longest side, opposite the right angle. This is your new base.

3

Sanity check the size

The answer must come out shorter than either leg. If it does not, something went in the wrong slot.

A worked example

Take a 3, 4, 5 triangle. Multiply the legs: 3 times 4 is 12. Divide by the hypotenuse: 12 divided by 5 is 2.4. The height onto the hypotenuse is 2.4 units.

Check it a second way. The area is half of 3 times 4, which is 6. Doubling that and dividing by the hypotenuse gives 12 divided by 5, which is 2.4 again. And notice the answer is smaller than 3, the shorter leg, which is exactly what you would expect once you see that the hypotenuse is the longest side to stand on.

A 3, 4, 5 right triangle with a dashed altitude drawn from the right angle onto the hypotenuse, labelled 2.4, beside two routes to that answer and the hypotenuse split into 1.8 and 3.2.
The dashed line is the only one of the four lengths in this picture that is not a side of the triangle. That is the whole difficulty with this particular height.

The legs are heights for each other, but not for the hypotenuse

Every side of a right triangle has its own height, and two of the three happen to be sides themselves. The third is not, and reaching for a leg because it worked twice already is the error this whole page exists for.

base 4givesheight is the leg of 3area 6

the legs are already perpendicular, so each is the other height

base 5givesheight is a leg of 3 or 4wrong

neither leg is square to the hypotenuse, so neither can be its height

The two pieces the hypotenuse splits into

Dropping the altitude cuts the hypotenuse into two segments, and on a 3, 4, 5 triangle those come out as 1.8 and 3.2. They add to 5, as they must. What is less obvious is that the height is the square root of their product: 1.8 times 3.2 is 5.76, and the square root of that is 2.4.

This is the geometric mean relation, and it gives you a second route to the answer when the segments are what you know rather than the legs. It also produces three similar triangles from one, since the altitude splits the original into two smaller right triangles that both have the same angles as the whole.

Mistakes that catch people out

  • Using a leg as the height onto the hypotenuse. Neither leg is perpendicular to the hypotenuse, so neither can serve as its height.
  • Dividing by a leg instead of the hypotenuse. The divisor is whichever side you are standing the triangle on, which here is the longest one.
  • Getting an answer larger than a leg. Impossible for this height. If it happens, the legs and hypotenuse went into the wrong slots.
  • Forgetting the product of the legs is twice the area. There is no half in h = ab / c, because the two halves cancelled during the derivation.
  • Applying it to a triangle with no right angle. The formula depends on the legs being perpendicular. Use double the area over the base instead.

Practice questions

  1. A 6, 8, 10 triangle. Find the height onto the hypotenuse.
  2. A 5, 12, 13 triangle. Find the height onto the hypotenuse.
  3. A 9, 12, 15 triangle. Find the height onto the hypotenuse.
  4. A right triangle with legs of 3 and 4. Find the two pieces the hypotenuse splits into.
  5. A right triangle with area 30 and hypotenuse 13. Find the height onto the hypotenuse.
Show the answers
  1. h = 6 x 8 / 10 = 48 / 10 = 4.8 units
  2. h = 5 x 12 / 13 = 60 / 13 = 4.62 units
  3. h = 9 x 12 / 15 = 108 / 15 = 7.2 units
  4. 1.8 and 3.2, which add to 5, and √(1.8 x 3.2) = 2.4 confirms the height
  5. h = 2 x 30 / 13 = 60 / 13 = 4.62 units

Common questions

What is the height of a right triangle to the hypotenuse?

Multiply the two legs and divide by the hypotenuse, so h = ab / c. On a 3, 4, 5 triangle that gives 12 divided by 5, which is 2.4.

Why is the height not one of the legs?

The legs are the heights for each other, because they already meet at a right angle. Once the hypotenuse becomes the base, the perpendicular to it is a different line entirely.

Should the answer be smaller than the legs?

Yes, always. The hypotenuse is the longest side, so standing the triangle on it gives the shortest of the three heights.

Can I use the area instead?

Yes. Double the area and divide by the hypotenuse. It is the same calculation, since twice the area is the product of the two legs.

What are the two pieces the hypotenuse splits into?

The altitude divides the hypotenuse into two segments. On a 3, 4, 5 triangle they are 1.8 and 3.2, and the height is the square root of their product.

Does this work on triangles that are not right angled?

No. For any other triangle, double the area and divide by whichever side you are using as the base. That method works everywhere, including here.

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