Find the Height of a Triangle with the Pythagorean Theorem

To find the height of an isosceles triangle, draw the altitude from the top corner to the base. It splits the triangle into two matching right triangles, each with hypotenuse s (the equal side) and leg b over 2 (half the base), so h = √s² − (b/2)². For an equilateral triangle with side s, h = √3 over 2s.

What "height" means

The height (or altitude) of a triangle is the perpendicular distance from one corner to the line containing the opposite side, which is then called the base. Every triangle has three heights, one for each choice of base. The height is what you need for the area formula:

Because the height meets the base at 90°, it always creates right triangles, and that is where the Pythagorean theorem comes in. In a right triangle the height is easy, because the two legs are already heights of each other. For any other triangle you have to work it out.

Height of an isosceles triangle

An isosceles triangle has two equal sides, s, and a third side called the base, b. Draw the altitude from the top corner, where the equal sides meet. By symmetry it lands exactly in the middle of the base, so it splits the isosceles triangle into two identical right triangles. (Formally, the two halves share the altitude and have equal hypotenuses s, so they are congruent by the hypotenuse-leg rule.)

Each half has hypotenuse s, one leg b over 2 and the other leg h. The theorem says h² + (b over 2)² = s². Solve for h, exactly as you would to find a missing leg:

The same split appears on every gable roof. A roof that spans 8 m with rafters 5 m long rises √5² − 4² = 3 m above the walls. The rafter length calculator does this, and the reverse, for any roof pitch.

Height of an equilateral triangle

An equilateral triangle has all three sides equal to s, so it is isosceles with base s. Half the base is s over 2, and the formula above becomes:

So the height of an equilateral triangle is always about 86.6% of its side. Each half is a 30-60-90 triangle: the altitude splits the 60° top angle into two 30° angles, and the base angles stay 60°.

For quick answers with any side or height, the special right triangle calculator handles 30-60-90 triangles directly.

Area formulas that follow

Once you have the height, the area is 1 over 2 × base × height. Putting the height formulas into it gives two shortcuts that let you skip finding h altogether.

Equilateral triangle. The base is s and the height is √3 over 2s:

For side 6 the area is √3 over 4 · 36 = 9√3 ≈ 15.59. For side 10 it is 25√3 ≈ 43.30.

Isosceles triangle. With equal sides s and base b:

Check it on Example 1: 10 over 4√4 · 169 − 100 = 2.5 · √576 = 2.5 · 24 = 60, which matches.

Height of any triangle from its three sides

A scalene triangle has no equal sides, so the altitude does not land in the middle of the base. You can still find the height with the Pythagorean theorem; it just takes two equations instead of one. Take the famous 13-14-15 triangle standing on its side of length 14.

1. Name the unknowns. Let the altitude have length h and land a distance x from the left end, so the right-hand piece of the base is 14 − x.
2. Write the theorem for both right triangles. Left: x² + h² = 13² = 169. Right: (14 − x)² + h² = 15² = 225.
3. Subtract to remove h. (14 − x)² − x² = 225 − 169, which simplifies to 196 − 28x = 56, so x = 5.
4. Find h. h = √169 − 25 = √144 = 12, and the area is 1 over 2 · 14 · 12 = 84.

In general, for base b and other sides p (on the left) and q (on the right), the same steps give x = b² + p² − q² over 2b and h = √p² − x². If x comes out negative or larger than b, the foot of the altitude lands outside the base: the triangle is obtuse, as explained on the acute, right or obtuse page. (Heron's formula gives the area from three sides directly, but the method above also tells you the height and where it falls.)

The height drawn to the hypotenuse of a right triangle has its own special properties, covered on the geometric mean theorem page.

Slant height of cones and pyramids

The same right-triangle split works in three dimensions. The height of a cone or pyramid goes straight down from the tip to the center of the base. The slant height runs down the outside surface: to the edge of the base on a cone, or to the midpoint of a base edge on a pyramid. Height, slant height and a distance across the base form a right triangle with the slant height as hypotenuse.

The distance from the top of a pyramid to a base corner takes one more step, because the corner is half a diagonal away from the center. That two-step method is explained on the Pythagorean theorem in 3D page.

Common mistakes

• Using the whole base instead of half. For the 13-13-10 triangle, √13² − 10² = √69 ≈ 8.31 is wrong; the leg of each right triangle is 5, not 10.
• Assuming the altitude bisects every base. It lands in the middle only when the two other sides are equal. For a scalene triangle, use the two-equation method.
• Adding instead of subtracting. The equal side s is the hypotenuse of each half, so the height is √s² − (b/2)². A height can never be longer than s.
• Mixing up height and slant height. The slant height is always the longer of the two, because it is the hypotenuse.

More slips like these are on the common mistakes page.