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

The two-formula chain is unavoidable here — you can’t get area directly from b and s without computing h first.

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 — pizza slice:

A pizza cut into 8 slices. The pizza is 14 in diameter, so each slice is an isosceles triangle with two equal sides of 7 in and a base equal to the slice arc length at the crust (technically curved, but we’ll approximate it as a straight line).

If each slice spans 45° (360°/8) of the full circle, the base length ≈ 2 × 7 × sin(22.5°) ≈ 5.36 in. Then h = √(49 − 2.875) ≈ 6.79 in. Area ≈ 0.5 × 5.36 × 6.79 ≈ 18.20 sq in per slice. Eight slices × 18.20 = 145.6 sq in vs the real pizza area π × 49 ≈ 153.9 sq in — the missing 8 sq in is the curved-edge part each slice loses to the straight-base approximation.

Bunting / 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 a soffit and verge trim along the slanted edges, each slanted edge is one leg of the isosceles triangle. For trim measurement, you need the leg length, not the height — buy lumber long enough to span s, not h.

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.