Isosceles Triangle Calculator

What is an Isosceles Triangle?

An isosceles triangle is a triangle with at least two equal sides. The two equal sides are called the legs, and the third side is called the base. The two angles opposite the equal legs (the base angles) are also equal — this is known as the Base Angles Theorem. Isosceles triangles appear frequently in geometry, architecture, and engineering because of their symmetry. Explore more triangle types with our Triangle Calculator and Equilateral Triangle Calculator.

The altitude from the apex (the vertex where the two equal sides meet) to the base is also the perpendicular bisector of the base, the angle bisector of the apex angle, and the median from the apex. Cutting along this altitude splits the isosceles triangle into two congruent right triangles, which is why the Pythagorean theorem is central to calculating isosceles triangle properties.

Formulas and Calculations

For an isosceles triangle with equal legs (a) and base (b):

Height (h): h = √(a² - (b/2)²)

The height is the perpendicular distance from the apex to the base. It is derived from the Pythagorean theorem applied to the right triangle formed by dropping the altitude.

Area (A): A = (1/2) × b × h = (b/4) × √(4a² - b²)

The area is half the product of the base and the height.

Perimeter (P): P = 2a + b

The perimeter is the sum of the two equal legs and the base.

Apex Angle (θ): θ = 2 × arctan(b / 2h)

The apex angle is at the vertex where the two equal legs meet. Each base angle equals (180° - θ) / 2.

How to Use the Isosceles Triangle Calculator

Enter the leg length (a) and base length (b) of the isosceles triangle. The calculator instantly computes the height, area, perimeter, apex angle, and base angles. The triangle inequality requires that the base be less than twice the leg length (b < 2a), and the calculator will warn you if this condition is not met.

All calculations are performed entirely in your browser with no server requests, ensuring instant feedback and privacy.