Isosceles Triangle Calculator

An isosceles triangle is a triangle with exactly two sides of equal length, called legs. The third side is called the base. The two angles opposite the equal legs are also equal and are called base angles, while the angle between the two legs is called the vertex angle.

The isosceles triangle theorem states that if two sides of a triangle are equal, then the angles opposite those sides are also equal. Conversely, if two angles of a triangle are equal, then the sides opposite those angles are equal. This means every isosceles triangle has a line of symmetry along its vertex height.

Use the formula: Area = (b/4) × √(4a² − b²), where 'a' is the leg length and 'b' is the base. This derives from computing the height h = √(a² − (b/2)²) and then applying Area = (1/2) × b × h.

The perimeter is simply P = 2a + b, where 'a' is the length of each equal leg and 'b' is the base. Because two sides are equal, you only need these two measurements.

Using the formula Area = (b/4) × √(4a² − b²) with a = 4 and b = 4: Area = (4/4) × √(64 − 16) = 1 × √48 ≈ 6.928 square units.

If you know the leg (a) and base (b), the base angle α = arccos(b / (2a)) and the vertex angle β = 180° − 2α. Alternatively, if you know one angle, the others follow because all three must sum to 180°.

The inradius (r) is the radius of the largest circle that fits inside the triangle: r = Area / s, where s is the semi-perimeter. The circumradius (R) is the radius of the circle passing through all three vertices: R = (a × a × b) / (4 × Area). Both values are computed automatically by this calculator.

A golden triangle is a special isosceles triangle where the ratio of the leg to the base equals the golden ratio φ ≈ 1.618. Its base angles are each 72° and the vertex angle is 36°. It appears frequently in art, architecture, and geometry.