How do I solve an isosceles triangle?
Pick the two measures you know, such as the leg and the base. The isosceles triangle calculator finds the other sides, the height, the vertex and base angles, the area and the perimeter, and draws the triangle to scale.
- Area
- 12
An isosceles triangle with legs 5 and base 6 has an area of 12 and a perimeter of 16.
- Leg (a)
- 5
- Base (b)
- 6
- Height (h)
- 4
- Perimeter
- 16
- Vertex angle (°)
- 73.73979529
- Base angles (°)
- 53.13010235
- Height to a leg
- 4.8
Area: 12. An isosceles triangle with legs 5 and base 6 has an area of 12 and a perimeter of 16.
What does the triangle look like?
How to calculate
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.
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.
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).
- 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
Each example is checked against the calculator on every build.
- I know Leg and base, Leg (a) 5, Base (b) 6 gives Height (h) 4, Area 12, Perimeter 16, Vertex angle (°) 73.739795, Base angles (°) 53.130102, Height to a leg 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)
- I know Leg and vertex angle, Leg (a) 10, Vertex angle 60 gives Base (b) 10, Height (h) 8.660254, Area 43.30127, Base angles (°) 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)
- I know Base and height, Base (b) 8, Height (h) 3 gives Leg (a) 5, Area 12, Perimeter 18, Vertex angle (°) 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)
- I know Base and base angle, Base (b) 6, Base angle 45 gives Height (h) 3, Leg (a) 4.242641, Vertex angle (°) 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)
- I know Leg and height, Leg (a) 13, Height (h) 12 gives Base (b) 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)
How it works
An isosceles triangle has two equal legs a and a base b. The height h from the vertex angle to the base meets the base at its middle, at a right angle. That gives two right triangles with legs w = b ÷ 2 and h and hypotenuse a.
Every choice is turned into w and h:
| I know | w (half the base) | h (height) |
|---|---|---|
| Leg a and base b | b ÷ 2 | √(a² − w²) |
| Base b and height h | b ÷ 2 | h |
| Leg a and height h | √(a² − h²) | h |
| Leg a and vertex angle V | a × sin(V ÷ 2) | a × cos(V ÷ 2) |
| Base b and base angle B | b ÷ 2 | w × tan B |
Then:
- Leg: a = √(w² + h²) (the typed leg when one was typed).
- Base: b = 2w. Perimeter: 2a + b.
- Area: w × h = ½ × b × h.
- Vertex angle: 2·atan2(w, h). Base angles: atan2(h, w) each, in degrees.
- Height to a leg: 2 × area ÷ a.
Rules
- Lengths are from 10⁻¹² to 10¹², in any one unit; the area is in that unit squared.
- The vertex angle is more than 0° and less than 180°; a base angle is more than 0° and less than 90°.
- The base must be shorter than two legs, and the height shorter than the leg; otherwise there is no triangle.
Output format. a² − w² and a² − h² are computed exactly from the typed decimals, then the square root and the trigonometry use double precision; values show to 10 significant figures. Angles are in degrees (radians × 180 ÷ π).
Worked examples by hand
Legs 5, base 6. w = 3, h = √(25 − 9) = 4. Area 3 × 4 = 12, perimeter 10 + 6 = 16. Vertex angle 2·atan(3 ÷ 4) = 73.7398°, base angles atan(4 ÷ 3) = 53.1301°. Height to a leg 24 ÷ 5 = 4.8.
Legs 10, vertex angle 60°. w = 10 sin 30° = 5, so the base is 10; h = 10 cos 30° = 5√3 = 8.6603. Area 5 × 8.6603 = 43.3013. Base angles 60°: the triangle is equilateral.
Base 8, height 3. w = 4, a = √(16 + 9) = 5. Area 12, perimeter 18, vertex angle 2·atan(4 ÷ 3) = 106.2602°.
Base 6, base angle 45°. w = 3, h = 3 tan 45° = 3, a = 3√2 = 4.2426, vertex angle 90°, area 9.
Leg 13, height 12. w = √(169 − 144) = 5, base 10, area 60, perimeter 36.
Other questions people ask
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°.