Isosceles Triangle Calculator

An isosceles triangle is a triangle with at least two sides of equal length. The two equal sides are called legs, and the third side is called the base. The angles opposite the equal sides (base angles) are also equal.

The base angles are the two angles opposite the equal sides. In an isosceles triangle, the base angles are always equal. If the apex angle is θ, each base angle is (180° - θ)/2.

Use the Pythagorean theorem: h = √(l² - (b/2)²), where l is the leg length and b is the base. The altitude from the apex bisects the base, creating two right triangles.

Use the Pythagorean theorem: l = √(h² + (b/2)²), where h is the height and b is the base. The leg is the hypotenuse of the right triangle formed by the altitude.

Use the Pythagorean theorem: b = 2 × √(l² - h²), where l is the leg and h is the height. Multiply the half-base by 2 to get the full base.

The apex angle is the angle between the two equal legs. It is opposite the base. If the base angles are α, the apex angle is 180° - 2α.

First find the half-base angle: α = arctan(h / (b/2)). Then the apex angle is 180° - 2α. Alternatively, use the law of cosines: cos(A) = (l² + l² - b²)/(2l²).

An equilateral triangle is a special case of an isosceles triangle where all three sides are equal (not just two). Every equilateral triangle is isosceles, but not every isosceles triangle is equilateral.

Yes. A right isosceles triangle has a 90° apex angle and two 45° base angles. The legs are equal, and the hypotenuse is leg × √2.

Perimeter = base + 2 × leg. For example, if the base is 10 and each leg is 7, the perimeter is 10 + 7 + 7 = 24.

Leg = (Perimeter - base) / 2. For example, if the perimeter is 24 and the base is 10, each leg is (24 - 10)/2 = 7.

Area = (1/2) × l² × sin(A), where l is the leg length and A is the apex angle in degrees. This is derived from A = (1/2)ab sin(C) for any triangle.

The inradius r = A/s, where A is the area and s is the semi-perimeter. For an isosceles triangle, r = (b × h) / (2l + b).

The circumradius R = (l² × b) / (4A), where l is the leg, b is the base, and A is the area. This can also be written as R = l² / (2h).

Isosceles triangles are common in roof trusses, gables, pediments, and A-frame structures. The equal legs distribute weight symmetrically, providing structural stability.

An isosceles triangle has one line of symmetry: the altitude from the apex to the base. This line bisects the apex angle, the base, and is perpendicular to the base.

Yes. For the apex angle A: cos(A) = (l² + l² - b²)/(2l²) = 1 - b²/(2l²). For a base angle B: cos(B) = (l² + b² - l²)/(2lb) = b/(2l).

An isosceles triangle can be split into two right triangles by the altitude. If the apex angle is 120°, each half is a 30-60-90 triangle. This is useful for solving problems involving 120° isosceles triangles.

Base = 2 × Area / Height. For example, if the area is 24 and the height is 6, the base is 2 × 24 / 6 = 8.

For base angle α: sin(α) = h/l, cos(α) = (b/2)/l = b/(2l), tan(α) = h/(b/2) = 2h/b. These ratios follow directly from the right triangle formed by the altitude.

The Isosceles Triangle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Isosceles Triangle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.