How isosceles triangles are calculated
The height splits the triangle into two matching right triangles.
Base angle = (180° − apex) ÷ 2; apex = 180° − 2 × base angle; height = √(leg² − (base ÷ 2)²); area = base × height ÷ 2; perimeter = 2 × leg + base.
- Subtract the apex angle from 180°, then divide by 2 to get each base angle. In radians, use π instead of 180°.
- For a known base angle, subtract twice that angle from 180° or π radians.
- With leg and base, use half the base and the Pythagorean theorem to find height.
- With leg and apex, base = 2 × leg × sin(apex ÷ 2) and height = leg × cos(apex ÷ 2).
- With base and height, leg = √((base ÷ 2)² + height²). Area is half base times height; perimeter is the sum of all three sides.
Worked example: Leg 5 and base 6 give height √(5² − 3²) = 4, area 6 × 4 ÷ 2 = 12 and perimeter 5 + 5 + 6 = 16. Each base angle is 53.13° and the apex is 73.74°.
Common isosceles triangle examples
| I know the | Apex angle | Base angle | Leg (equal side) | Base | Apex angle | Base angle 1 | Area |
|---|---|---|---|---|---|---|---|
| Apex angle | 40 ° | — | — | — | 40° | 70° | Not determined from angles alone |
| Apex angle | 90 ° | — | — | — | 90° | 45° | Not determined from angles alone |
| Base angle | — | 70 ° | — | — | 40° | 70° | Not determined from angles alone |
| Leg and base | — | — | 5 units | 6 units | 73.74° | 53.13° | 12 square units |
| Leg and apex angle | 60 ° | — | 10 units | — | 60° | 60° | 43.3013 square units |
Swipe the table sideways to see the rest.
Lengths use the same arbitrary unit. Angles are in degrees.
Things to consider
- The legs are the two equal sides, not the base.
- The base must be shorter than twice a leg.
- A single angle cannot determine lengths or area.
- An equilateral triangle is also isosceles because at least two sides are equal.
- Use the Area Calculator to compare areas across different shapes.
Frequently asked questions
What is an isosceles triangle?
It is a triangle with at least two equal sides. These equal sides are the legs. The angles opposite them are the two equal base angles. An equilateral triangle is a special case with three equal sides.
How do I find the base angles from the apex?
Subtract the apex from 180°, then divide by 2. An apex of 40° leaves 140° for the two equal base angles, so each is 70°. In radians, subtract the apex from π and divide by 2.
Why are the base angles equal?
Equal sides have equal opposite angles. The height from the apex divides an isosceles triangle into two congruent right triangles, which have matching base angles.
How do I find the height?
The height bisects the base. Using the Pythagorean theorem, height = √(leg² − (base ÷ 2)²). For leg 5 and base 6, the height is 4. See the Pythagorean Theorem Calculator: Find the Hypotenuse for other right-triangle side calculations.
Can an isosceles triangle be right-angled or obtuse?
Yes. An apex of 90° gives base angles of 45°, making a right isosceles triangle. See the 45-45-90 Triangle Calculator for that special case. An apex above 90° makes the triangle obtuse, with both base angles below 45°.
Results are estimates for general information only, not financial, medical or professional advice.