Find the base and apex angles of an isosceles triangle from its base and equal-side (leg) length.
How it works
Splitting the triangle down the middle creates a right triangle: height = √(leg² – (base/2)²), then the base angle is arctan(height / (base/2)), and the apex angle is 180° minus twice the base angle. A base of 6 with legs of 5 gives base angles of 53.13° and an apex angle of 73.74°.
What this does not include
This is distinct from this site’s isosceles triangle area calculator, which reports area and height from the same two inputs but not the angles.
How to use this calculator
- Enter the base length.
- Enter the equal side (leg) length.
A worked example
An isosceles triangle with base 6, equal legs of 5: base angles = 53.1301° each, apex angle = 73.7398°.
Base 10, legs 13: base angles = 67.3801°, apex angle = 45.2397°.
What the variables mean
| Variable | Meaning |
|---|---|
| Base | Length of the unequal (base) side |
| Leg | Length of either of the two equal sides |
Edge cases worth knowing
The two base angles are always equal to each other — the defining property of an isosceles triangle, so only one needs to be calculated and the other is automatically known.
Not every base-and-leg combination forms a valid triangle. A base of 10 with legs of only 4 fails the triangle inequality — the two legs can’t reach across a base that long — so the calculator declines to show a result.
Why do the two base angles have to be equal?
By definition, an isosceles triangle has two equal sides, and the angles opposite those equal sides are always equal to each other — that’s a basic property of isosceles triangles.
What if the legs are too short to reach across half the base?
No valid triangle exists in that case — the two legs simply can’t meet, so this calculator returns no result rather than an invalid negative-square-root answer.
Do the three angles always add up to 180°?
Yes — that’s true of every triangle, and this calculator’s formula guarantees it since the apex angle is defined as 180° minus the two base angles.
