Isosceles Triangle Calculator — Side, Angle, Area & Perimeter Solver
Solve any isosceles triangle from the measurements you have. Pick one of the four modes below — the solver returns the equal sides, the base, the apex angle, the base angles, the height, the area, and the perimeter.
| Equal sides a (legs) | |
|---|---|
| Base b | |
| Apex angle | |
| Base angles (each) | |
| Height (altitude) h | |
| Area | |
| Perimeter |
Formulas used
Worked examples
Example 1 — equal sides a = 5, base b = 6: height = 4.0, area = 12.0, perimeter = 16.0, base angle = 53.13010235415598°, apex angle = 73.73979529168804°.
Example 2 — equal side a = 10, apex 40°: base = 6.840402866513374, height = 9.396926207859082, area = 32.139380484326956.
Example 3 — base b = 20, base angle 65° (side solver): equal side = 23.662015831524986, area = 214.45069205095587.
Example 4 — a = 13, b = 10: height = 12.0, area = 60.0, perimeter = 36.0, base angle = 67.38013505195957°, apex = 45.23972989608085°.
Also searched as: isosceles triangle angle calculator, isosceles triangle area calculator, isosceles triangle side calculator, isosceles triangle side length calculator.
Boundary note: this is a specialized isosceles solver — it recovers missing sides and both angles from partial data (side + apex angle, base + base angle). A general triangle area calculator handles any triangle but cannot do that. For non-isosceles triangles, use the general tool.
Frequently asked questions
What is an isosceles triangle?
An isosceles triangle is a triangle with two equal sides (the legs, a) and a base (b). The two angles opposite the equal sides — the base angles — are always equal, and the angle between the two equal sides is the apex (vertex) angle. Since the three angles sum to 180°, each base angle equals (180° - apex) / 2.
How do you calculate the area of an isosceles triangle?
First find the height (altitude) with h = √(a² - (b/2)²), then area = b × h / 2. Example: equal sides a = 5 and base b = 6 give h = √(25 - 9) = 4.0, so the area is 6 × 4.0 / 2 = 12.0 square units.
How do you find the base angles of an isosceles triangle?
Use base = (180° - apex) / 2. For example, an apex angle of 40° gives base angles of (180 - 40) / 2 = 70°. Conversely, from the sides: for a = 5, b = 6, the base angle is 53.13010235415598° and the apex angle is 73.73979529168804°.
How do I find a missing side of an isosceles triangle?
This page is a complete isosceles triangle side calculator. From the base and a base angle: a = (b/2) / cos(base). Example: base b = 20 and base angle 65° give equal sides a = 23.662015831524986. From the equal side and the apex angle: b = 2 × a × sin(apex/2) — for a = 10 and apex 40°, b = 6.840402866513374.
How do you find the side length of an isosceles triangle from the apex angle?
Use b = 2 × a × sin(apex/2). With equal side a = 10 and apex angle 40°, the base is 2 × 10 × sin(20°) = 6.840402866513374. With a = 12 and apex 100°, the base is 18.385066634855473.
How do you find the height (altitude) of an isosceles triangle?
Use h = √(a² - (b/2)²). For a = 5, b = 6: h = √(25 - 9) = 4.0. For a = 13, b = 10: h = √(169 - 25) = 12.0. Alternatively, from height and base you can recover the equal side: a = √((b/2)² + h²) — height 4 and base 6 give a = 5.0.
How do you find the apex angle from the sides?
Use apex = arccos(1 - b²/(2a²)). For a = 5, b = 6: apex = arccos(1 - 36/50) = 73.73979529168804°, and each base angle is 53.13010235415598°.
What is the perimeter of an isosceles triangle?
P = 2a + b. For a = 5, b = 6 the perimeter is 16.0. For a = 13, b = 10 it is 36.0.
Can an isosceles triangle be equilateral too?
Yes — an equilateral triangle is a special isosceles triangle with all three sides equal. Enter base 12 and base angle 60° and the solver returns equal sides ≈ 11.999999999999998, apex ≈ 60.00000000000002°, height 10.39230484541326, and area 62.35382907247956.
What inputs does this isosceles triangle solver accept?
Four modes: (1) the two equal sides and the base, (2) one equal side and the apex angle, (3) the base and a base angle — this is the built-in isosceles triangle side solver, and (4) the height and the base. Every mode returns all sides, both angles, the height, the area, and the perimeter.
How is this different from a general triangle area calculator?
A general triangle area calculator handles any triangle from base and height or three sides, but it does not exploit the symmetry of an isosceles triangle. This solver is specialized: it recovers missing sides and both angles from partial information (for example side + apex angle, or base + base angle), which a general area tool cannot do.
What makes an input invalid for an isosceles triangle?
The triangle inequality requires 2a > b (for example a = 5, b = 12 is rejected), the apex angle must be strictly between 0° and 180°, each base angle strictly between 0° and 90°, and all lengths positive. The solver shows an error message for any invalid combination.
Free tool by NRTechStudio