Why is the base limited to twice the equal side?

Two sides of length a can reach at most a distance of 2a apart when laid flat, so a larger base cannot close into a triangle. Exactly 2a collapses it into a straight line.

How are the apex and base angles related?

The two base angles are equal and the apex angle is 180° minus twice one base angle. A tall, narrow triangle has a small apex angle and base angles close to 90°.

Which formulas does the calculator use for angles?

It uses cos(apex / 2) = (base / 2) / equal side, which follows from the right triangle formed by the altitude.

📐 Isosceles Triangle Calculator

Calculate sides, area and angles of an isosceles triangle.

Equal sides length (a) Base length (b)

What is an Isosceles Triangle?

A triangle with two equal sides. The angles opposite the equal sides (base angles) are also equal. Apex angle (α) and base angle (β): α + 2β = 180°

Height Formula

h = √(a² − (b/2)²)

a: equal sides, b: base

Area Formula

S = (b × h) / 2

Apex Angle

α = 2 × arcsin(b / (2a))

Base Angle

β = (180° − α) / 2

Perimeter

L = 2a + b

Properties of an Isosceles Triangle

The perpendicular from the apex angle to the base bisects the base perpendicularly. That is, the foot of the perpendicular is the midpoint of the base.

Worked examples

Equal sides of 5 with a base of 6 give a height of √(25 − 9) = 4, an area of ½ × 6 × 4 = 12, a perimeter of 16, an apex angle of about 73.7398° and base angles of about 53.1301°.

Equal sides of 10 with a base of 10 make an equilateral triangle: height 5√3 ≈ 8.6603, area 25√3 ≈ 43.3013, perimeter 30 and all angles 60°.

What the result assumes

The altitude drops exactly onto the midpoint of the base, so the height splits the triangle into two congruent right triangles. The base cannot exceed the sum of the two equal sides, and the apex angle is capped at 180° as the base approaches zero. Angles are shown in degrees; the ratio height to half-base drives them.

Frequently asked questions

Why is the base limited to twice the equal side?: Two sides of length a can reach at most a distance of 2a apart when laid flat, so a larger base cannot close into a triangle. Exactly 2a collapses it into a straight line.
How are the apex and base angles related?: The two base angles are equal and the apex angle is 180° minus twice one base angle. A tall, narrow triangle has a small apex angle and base angles close to 90°.
Which formulas does the calculator use for angles?: It uses cos(apex / 2) = (base / 2) / equal side, which follows from the right triangle formed by the altitude.