# How do I solve an isosceles triangle?

Solves an isosceles triangle from its leg and base, base and height, leg and height, leg and vertex angle, or base and base angle: every side, the height, the angles, the area and the perimeter.

- Page: https://www.acalculator.org/math/isosceles-triangle-calculator
- JSON spec: https://www.acalculator.org/math/isosceles-triangle-calculator.json
- Version: b8bbe3e3bc4d

## Default answer

Example with the default inputs (I know Leg and base, Leg (a) 5, Base (b) 6): An isosceles triangle with legs 5 and base 6 has an area of 12 and a perimeter of 16.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| known | I know | Which two measures of the triangle you know. |
| a | Leg (a) | The length of each of the two equal sides. |
| b | Base (b) | The length of the third, unequal side. |
| h | Height (h) | The distance from the vertex angle straight down to the base. |
| v | Vertex angle | The angle between the two equal legs, in degrees. |
| ba | Base angle | Each of the two equal angles at the base, in degrees. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| area | Area | Half the base times the height. |
| leg | Leg (a) | Each of the two equal sides. |
| base | Base (b) | The unequal side. |
| height | Height (h) | The height to the base. |
| perimeter | Perimeter | Two legs plus the base. |
| vertex | Vertex angle (°) | The angle between the legs, in degrees. |
| baseAngle | Base angles (°) | Each of the two equal base angles, in degrees. |
| legHeight | Height to a leg | The distance from a base corner straight to the opposite leg: 2 × area ÷ leg. |

## Method

With half the base w = b ÷ 2 and the height h: a² = h² + w², area = w × h, perimeter = 2a + b, vertex angle = 2·atan(w ÷ h), base angle = atan(h ÷ w).

## Assumptions

- The two legs are equal; the base is the third side.
- Lengths are in any one unit, and the area is in that unit squared. Angles are in degrees.

## Worked examples

1. known = lb, a = 5, b = 6 gives height = 4, area = 12, perimeter = 16, vertex = 73.739795, baseAngle = 53.130102, legHeight = 4.8. Source: OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05); OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh), https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids (retrieved 2026-10-05).
2. known = lv, a = 10, v = 60 gives base = 10, height = 8.660254, area = 43.30127, baseAngle = 60. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sin, cos and tan as side ratios), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05).
3. known = bh, b = 8, h = 3 gives leg = 5, area = 12, perimeter = 18, vertex = 106.260205. Source: OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05).
4. known = ba, b = 6, ba = 45 gives height = 3, leg = 4.242641, vertex = 90, area = 9. Source: OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sin, cos and tan as side ratios), https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05).
5. known = lh, a = 13, h = 12 gives base = 10, area = 60, perimeter = 36. Source: OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; the angles of a triangle add to 180°), https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05).

## FAQ

### How do I find the height of an isosceles triangle?

The height to the base cuts the triangle into two right triangles, each with hypotenuse a (the leg) and one leg b ÷ 2. So h = √(a² − (b ÷ 2)²). Legs of 5 and a base of 6 give h = √(25 − 9) = 4.

### How do I find the area of an isosceles triangle?

Use A = ½ × base × height. With legs 5 and base 6, the height is 4, so A = ½ × 6 × 4 = 12.

### How do I find the angles of an isosceles triangle?

The two base angles are equal. Each is atan(h ÷ (b ÷ 2)), and the vertex angle is 180° minus twice that. Legs 5 and base 6 give base angles of 53.13° and a vertex angle of 73.74°.

### How do I find the base from the leg and the vertex angle?

b = 2a × sin(vertex angle ÷ 2), and the height is a × cos(vertex angle ÷ 2). Legs of 10 with a 60° vertex angle give a base of 10: the triangle is equilateral.

### What if the base is longer than two legs?

Then no triangle exists, because the two legs cannot reach across the base. The base must be shorter than 2a, and the height shorter than the leg.

### Is an equilateral triangle isosceles?

Yes. An isosceles triangle has at least two equal sides, so an equilateral triangle, with all three equal, is a special case. Its angles are all 60°.

## Sources

- OpenStax, Prealgebra 2e, §9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (the Pythagorean theorem; the angles of a triangle add to 180°), CC BY 4.0. https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem (retrieved 2026-10-05)
- OpenStax, Prealgebra 2e, §9.4 Use Properties of Rectangles, Triangles, and Trapezoids (the area of a triangle, A = ½bh), CC BY 4.0. https://openstax.org/books/prealgebra-2e/pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids (retrieved 2026-10-05)
- OpenStax, Algebra and Trigonometry 2e, §7.2 Right Triangle Trigonometry (sine, cosine and tangent as ratios of sides), CC BY 4.0. https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-2-right-triangle-trigonometry (retrieved 2026-10-05)