What this calculator does
An isosceles triangle has two sides of equal length, the legs, meeting at an apex above a third side, the base. That symmetry means the whole shape is fixed by just two measurements: the length of the equal sides and the length of the base.
This isosceles triangle calculator takes those two measurements and returns the area, perimeter, the height dropped from the apex to the base, and all three angles, including the apex angle sitting between the two equal sides.
The formula
The height from the apex to the base splits the triangle into two identical right triangles, so h = √(leg² − (base/2)²) by Pythagoras. Area follows as ½ × base × height. The apex angle comes from the same right triangle: apex = 2 × asin((base/2) / leg), and the two equal base angles share what is left of 180°.
| Term | Meaning |
|---|---|
| Leg | One of the two equal sides of the isosceles triangle. |
| Base | The third, unequal side, opposite the apex. |
| Apex angle | The angle between the two equal legs, at the top of the triangle. |
The inputs explained
| Field | What to enter |
|---|---|
| Equal side length (leg) | The length of either of the two equal sides. |
| Base | The length of the base, the unequal third side. |
When to use it
A roof truss or gable end
Many roof trusses and gable ends are built as an isosceles triangle, symmetric about the centre, so the equal-side length and base are usually the two measurements already on the plan.
A road sign or pennant shape
Warning signs, bunting and pennant flags are frequently isosceles triangles, and the area figure here gives the material needed to cut one out.
A school geometry problem
An isosceles triangle formula question that gives the two equal sides and the base is solved directly here, without needing to draw the perpendicular height by hand first.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does area change as the base widens?
A fixed leg length of 10, across a range of base lengths, showing how area and the apex angle both grow as the base widens.
| Base | Area | Apex angle (between the equal sides) |
|---|---|---|
| 4 | 19.596 | 23.07° |
| 8 | 36.661 | 47.16° |
| 12 | 48.000 | 73.74° |
| 16 | 48.000 | 106.26° |
| 19 | 29.664 | 143.61° |
How does area change as the equal sides lengthen?
A fixed base of 6, across a range of leg lengths, from just long enough to close the triangle up to a much taller shape.
| Equal side length (leg) | Area | Base angles (each) |
|---|---|---|
| 4 | 7.937 | 41.41° |
| 5 | 12.000 | 53.13° |
| 6 | 15.588 | 60.00° |
| 8 | 22.249 | 67.98° |
| 10 | 28.618 | 72.54° |
Questions
What is an isosceles triangle?
An isosceles triangle is a triangle with two sides of equal length, called the legs, and a third side, the base, which is usually a different length. The two angles opposite the equal legs are also equal.
What is the isosceles triangle formula for area?
Area = ½ × base × height, where the height is the perpendicular distance from the apex to the base. If only the leg and base lengths are known, the height is worked out first with Pythagoras: h = √(leg² − (base/2)²).
How do you find the angles of an isosceles triangle?
The two base angles are always equal to each other. Find the apex angle first from the leg and base lengths, then split the remaining 180° minus the apex angle equally between the two base angles.
Can an isosceles triangle also be equilateral?
Yes. An equilateral triangle, with all three sides equal, is a special case of an isosceles triangle where the base also happens to match the leg length, giving three 60° angles instead of a distinct apex angle.
For a triangle with three different known side lengths instead of two equal ones, see the triangle from three sides calculator. For a triangle with no two sides equal, see the scalene triangle calculator.