Isosceles Triangle Area Calculator

An isosceles triangle has two equal sides (the legs) and one different side (the base). The two angles opposite the equal sides are also equal.

From base b and the two equal legs s, the height h is:

h = √(s² − (b/2)²)

That’s the Pythagorean theorem applied to the right triangle formed by dropping a perpendicular from the apex to the base. The perpendicular bisects the base, so its foot is at distance b/2 from each base vertex.

Then area:

A = ½ × b × h

You can collapse the two steps into one if you prefer: A = (b/4) × √(4s² − b²) gives the same answer straight from b and s. Most people still compute h first, because the height is usually a number you need anyway.

A sanity-check shortcut. If b = s (so all three sides are equal), the triangle is equilateral and area = (√3/4) × s². If b approaches 2s, the triangle flattens to a line and area approaches zero. If b is small compared to s, the triangle is tall and skinny and most of its area sits near the base.

Worked example: a pizza slice

A 14 in pizza cut into 8 slices. Each slice is an isosceles triangle with two equal sides of 7 in (the radius) and a base that stands in for the curved crust.

Each slice spans 45°, so the base is 2 × 7 × sin(22.5°) = 5.358 in. Half the base is 2.679, and squaring that gives 7.176. h = √(49 − 7.176) = 6.467 in. Area = 0.5 × 5.358 × 6.467 = 17.32 sq in per slice.

Eight of those come to 138.6 sq in, against the real pizza area of π × 49 = 153.9. The missing 15.3 sq in, right on 10%, is the eight crescents of crust that the straight base cuts off. That is a real error, not a rounding one, and it is why pizza area is worth doing as a circle rather than as eight triangles.

Bunting and pennants

A simple party pennant is an isosceles triangle, usually with legs of 8 to 12 in and a base of 4 to 6 in. Area for an 8-in leg, 6-in base pennant: h = √(64 − 9) ≈ 7.42 in, area ≈ 22.25 sq in. 20 pennants = 445 sq in of fabric.

Roof gable verge

If the roof has soffit and verge trim along the slanted edges, each slanted edge is one leg of the isosceles triangle. For trim, you need the leg length, not the height. Buy lumber long enough to span s, never h, or the boards arrive short by exactly the amount the pitch adds.

When designing furniture, isosceles triangles show up in trestle table supports, roof rafters, and most “A-frame” structures. The geometry stays the same regardless of scale.