# Isosceles Triangle Calculator

Calculate isosceles triangle area, perimeter, height, base angles, and vertex angle from side lengths. Free online isosceles triangle calculator.

## What this calculates

Calculate all properties of an isosceles triangle from the equal side length and base length. This tool computes the area, perimeter, height, base angles, and vertex angle instantly.

## Inputs

- **Equal Side Length (a)** — min 0 — The length of each of the two equal sides.
- **Base Length (b)** — min 0 — The length of the base (the unequal side).

## Outputs

- **Area** — The area of the isosceles triangle.
- **Perimeter** — The total length of all three sides.
- **Height** — The perpendicular height from the base to the apex.
- **Base Angles** — Each of the two equal base angles.
- **Vertex Angle** — The angle at the apex between the two equal sides.

## Details

An isosceles triangle has two equal sides (called legs) and a third side (called the base). The two angles adjacent to the base are equal (base angles), and the angle between the two equal sides is the vertex angle.

Isosceles Triangle Formulas (equal sides = a, base = b):

- Height: h = sqrt(a^2 - (b/2)^2)

- Area: A = (b x h) / 2 = (b/4) x sqrt(4a^2 - b^2)

- Perimeter: P = 2a + b

- Base Angles: arccos(b / (2a))

- Vertex Angle: 180 - 2 x base angle

The altitude from the vertex to the base bisects both the base and the vertex angle, creating two congruent right triangles. This axis of symmetry is a key property of isosceles triangles.

Isosceles triangles appear in architecture (gable roofs, A-frame buildings), bridge design (truss structures), road signs, and decorative patterns. An equilateral triangle is a special case where the base equals the equal sides.

## Frequently Asked Questions

**Q: What is an isosceles triangle?**

A: An isosceles triangle has exactly two sides of equal length (called the legs) and a third side of different length (the base). The two angles opposite the equal sides (base angles) are also equal. If all three sides are equal, it is an equilateral triangle, which is a special case of isosceles.

**Q: How do I find the height of an isosceles triangle?**

A: The height (altitude from vertex to base) is h = sqrt(a^2 - (b/2)^2), where a is the equal side length and b is the base. This formula comes from the Pythagorean theorem applied to the right triangle formed by the altitude, half the base, and one equal side.

**Q: How do I find the angles of an isosceles triangle?**

A: The base angle is arccos(b / (2a)), where a is the equal side and b is the base. The vertex angle equals 180 minus twice the base angle. For example, if a = 10 and b = 12: base angle = arccos(6/10) = arccos(0.6) = 53.13 degrees, vertex angle = 180 - 106.26 = 73.74 degrees.

**Q: When is an isosceles triangle also a right triangle?**

A: An isosceles triangle is also a right triangle when the vertex angle is exactly 90 degrees, making both base angles 45 degrees. This is the famous 45-45-90 triangle. In this case, the base equals a x sqrt(2), where a is the equal side length.

---

Source: https://vastcalc.com/calculators/math/isosceles-triangle
Category: Math
Last updated: 2026-04-21