Isosceles Triangle Angles Calculator
Find the base angles and apex angle of an isosceles triangle — either from the two equal sides and the base, or from the apex angle and the base.
Known inputs
Interior angle at each end of the base (both equal)
- 1
Ratio b ÷ (2a)
6 ÷ (2 × 5) = 0.6In the right triangle formed by the altitude: cos(base angle) = (b/2) / a. - 2
Base angle
arccos(0.6) × 180 ÷ π = 53.1301
How does this calculator work?
For an isosceles triangle with equal sides a and base b: base angle = arccos(b / (2a)) and apex angle = 180° − 2 × base angle. Working backwards from the apex angle α and base b: base angle = (180° − α) / 2 and equal side a = b × sin(base angle) / sin(α).
Formula
How this is calculated
An isosceles triangle has two equal sides a and a base b. By symmetry the triangle is bisected by the altitude from the apex to the base, creating two congruent right triangles. In each right triangle the hypotenuse is a and the adjacent side is b/2, so the base angle satisfies cos(base angle) = (b/2) / a, giving base angle = arccos(b / (2a)). Because the three interior angles of any triangle sum to 180° and the two base angles are equal, the apex angle = 180° − 2 × base angle.
The "apex angle and base" mode works in reverse: given the apex angle α and the base b, each base angle is (180° − α) / 2. The equal side length is then found via the Law of Sines: a / sin(base angle) = b / sin(α), so a = b × sin(base angle) / sin(α). This recovers the full triangle from just two pieces of information — the apex angle and the base length.
All angles are reported in degrees. For any valid isosceles triangle the base angles are each in the range (0°, 90°) when the apex angle is acute, or they can exceed 90° when the apex angle is acute and the triangle is very "wide". The apex angle must be strictly between 0° and 180° for a non-degenerate triangle.
Frequently asked questions
The two base angles sit at the ends of the base, opposite the two equal sides. By the isosceles triangle theorem, the angles opposite equal sides are themselves equal. Geometrically, the triangle is bilaterally symmetric — reflecting it across the altitude from the apex maps one base angle onto the other.
No. Each base angle must be acute (less than 90°). If a base angle were 90°, the apex would be exactly at 0°, which is degenerate. And since both base angles are equal, two obtuse base angles would already sum to more than 180°, which is impossible for a triangle.
An equilateral triangle is a special isosceles triangle where a = b. All three angles equal 60°. Here arccos(b / (2a)) = arccos(1/2) = 60°, so each base angle is 60° and the apex angle is also 60°.
TG we-Calculate Editorial Team. (2026). Isosceles Triangle Angles Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/isosceles-triangle-angles-calculator
TG we-Calculate Editorial Team. "Isosceles Triangle Angles Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/isosceles-triangle-angles-calculator.
TG we-Calculate Editorial Team, "Isosceles Triangle Angles Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/isosceles-triangle-angles-calculator
@misc{wecalculate_isosceles_triangle_angles_calculator, title = {Isosceles Triangle Angles Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/isosceles-triangle-angles-calculator}}, year = {2026}, note = {TG we-Calculate} }
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