Isosceles Triangle Calculator
From the base and one equal leg, get the area, the height, the perimeter, and both kinds of angle — every number that describes an isosceles triangle.
Two inputs, five answers
Enter the base and the equal-side leg and the calculator returns the area (½ × base × height), the height, the perimeter, the two base angles, and the apex angle at once.
The leg must clear half the base
For a real triangle the leg has to be longer than half the base, otherwise the two equal sides cannot meet above the base — the calculator returns nothing for a flat, degenerate shape.
What is an isosceles triangle calculator?
Base and leg in, the whole triangle out
An isosceles triangle calculator turns two measurements — the base and one of the two equal sides (the leg) — into the numbers that describe the whole shape: the area, the height from the base up to the apex, the perimeter, the two equal base angles, and the single apex angle. Each one is fixed once you know the base and leg, because the two equal sides lock the symmetry: there is only one isosceles triangle with that base and that leg. That makes the two inputs all you need for roof gables, tent panels, pennant flags, logo shapes, and any geometry homework where a symmetric triangle shows up.
Enter the base and the equal leg in any length unit to get the area, height, perimeter, and all three angles instantly.
A few short formulas, all built from the base b and the equal leg a.
height = √(a² − (b / 2)²)The height drops a perpendicular from the apex to the middle of the base, splitting the triangle into two right triangles; by the Pythagorean theorem it is √(a² − (b/2)²). The area is then ½ × base × height. The perimeter is 2 × a + b. Each base angle is acos((b/2) / a), and the apex angle is whatever is left after the two equal base angles: 180 − 2 × base angle.
Suppose your isosceles triangle has a base of 6 and equal legs of 5.
Height
√(5² − 3²) = √(25 − 9) = √16 = 4 — the perpendicular from the apex (a 3-4-5 half-triangle).
Area and perimeter
½ × 6 × 4 = 12 square units, and 2 × 5 + 6 = 16 around the edge.
Angles
base angle = acos(3 / 5) ≈ 53.130102°, apex = 180 − 2 × 53.130102 ≈ 73.739795°.
The five outputs answer different practical questions. The area (12 square units for base 6, leg 5) is the flat space the triangle covers, handy for fabric, sail, or paint estimates. The height (4 here) is the perpendicular distance from the base up to the apex — the "rise" of a gable or the depth of a pennant — and is always shorter than the leg because the leg slants outward. The perimeter (16) is the distance all the way around, useful for trim or edging. The two base angles are equal by symmetry (about 53.13° here), and the apex angle (about 73.74°) sits at the top; the single most useful insight is that all three always sum to 180°, so apex = 180 − 2 × base angle. A taller, narrower triangle has a small apex angle and steep base angles; a short, wide one has a wide apex and shallow base angles.
The formulas are exact, but a couple of practical points are worth keeping in mind.
The leg must clear half the base, and keep units consistent
These formulas need a genuine isosceles triangle: the equal leg must be longer than half the base, so the two sides can meet above the base. If the leg is exactly or less than half the base, the shape collapses flat and the calculator returns no result. The base and leg are also unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a base and leg in centimetres give a height and perimeter in centimetres and an area in square centimetres, never a mix. The three angles are always reported in degrees.