How this instrument works
The vertex angle of an isosceles triangle is the figure sandwiched between its two equal legs, sitting opposite the base. Splitting the triangle exactly in half down its own height creates two identical right triangles, each holding exactly HALF that value, with half the base as the side opposite it and the leg itself as the hypotenuse — basic right-triangle trigonometry then gives the half-figure as asin((base⁄2) ⁄ a), and doubling that recovers the full vertex value.
This is a different output entirely from the combined isosceles-triangle page elsewhere on this site, which returns height, area, and perimeter from the same base-and-leg inputs but nothing about the corners at all. Here, that SAME pair of measurements instead reveals the vertex figure specifically — a genuinely different piece of information about the identical triangle.
This value depends only on the RATIO of the base to the leg, not on the triangle's absolute size — two isosceles triangles with different leg lengths but the same base-to-leg ratio will always share the exact same result, since scaling every length by the same factor leaves that ratio, and therefore the figure, completely unchanged.
- Enter the length of either equal leg into the Leg length (a) field.
- Enter the triangle's base into the Base field.
- Read Vertex angle: the sheet computes it using the half-figure right-triangle relationship.
Worked example — legs of 5, base of 6
An isosceles triangle has equal legs of 5 and a base of 6. The vertex figure is 2·asin(3⁄5) ≈ 73.74° — splitting the triangle in half creates a right triangle with hypotenuse 5 and opposite side 3 (half the base), giving a half-value of asin(0.6) ≈ 36.87°, doubled for the full result.
Equal legs of 5 with a base of 5 too makes this an equilateral triangle, and the result comes out to exactly 60° — confirming the formula correctly handles that special case. Scale everything up, legs of 10 with a base of 12 (the identical 6:5 ratio as the golden example), and the figure stays at exactly 73.74°, unchanged by the larger absolute size.
Questions
What is the vertex angle of an isosceles triangle?
It's the figure between the two equal legs, sitting opposite the base — sometimes called the apex angle. It's a different measure from the two equal base corners, which sit at the two ends of the base instead.
How is the vertex angle calculated from the base and leg?
vertex angle = 2·asin((base⁄2) ⁄ a) — splitting the triangle in half down its own height creates a right triangle holding exactly half the target value, with half the base opposite it and the leg as the hypotenuse; doubling that half-figure recovers the full result.
Does the result depend on the triangle's absolute size?
No — only on the RATIO of the base to the leg. Scaling every length of the triangle by the same factor leaves that ratio, and therefore the outcome, completely unchanged, since the asin function only sees (base⁄2)⁄a, not the individual lengths.
What is the maximum possible value here?
Approaching but never reaching 180°, as the base grows to nearly twice the leg length — right at base = 2·leg, the triangle flattens entirely into a straight line, the boundary this calculator's underlying check enforces.
How is this different from the isosceles-triangle combined page?
That page takes the same base-and-leg inputs but returns height, area, and perimeter — nothing about the corners. This page uses those identical two inputs to solve for the vertex figure specifically, a genuinely different piece of information about the same triangle.