An isosceles triangle has two equal sides, called the legs, and a third side called the base. The legs meet at the apex, and the two angles at the ends of the base are equal. Thanks to that symmetry, two measurements solve it, and this calculator accepts six different pairs.
How to use the isosceles triangle calculator
- Choose a pair in the I know the menu: Leg (a) and base (b), Base (b) and height (h), Leg (a) and height (h), Leg (a) and apex angle (γ), Base (b) and base angle (α), or Base (b) and apex angle (γ).
- Fill in the boxes that appear. The leg is either of the two equal sides; the height runs from the apex straight down to the base.
- Set the Units for lengths and the Angle unit (degrees by default, or radians).
- The headline is the area. The tape also lists the leg, base, height to base, base angles (α), apex angle (γ), perimeter, height to a leg, inradius, circumradius and the type: acute, right or obtuse isosceles, or equilateral.
If you know a leg and a base angle, convert first with γ = 180° − 2α, then use Leg (a) and apex angle (γ).
Isosceles triangle formulas
The height from the apex splits the triangle into two mirror-image right triangles with hypotenuse a and legs h and b/2. Every formula comes from that half-triangle:
The inradius is the area divided by half the perimeter, and the circumradius is a2 ÷ 2h.
Worked example: a gable end
A garage has a 24 ft wide gable wall, and the ridge sits 6 ft above the top of the wall (a 6-in-12 roof pitch). Choose Base (b) and height (h), enter b = 24 and h = 6, and set the units to ft.
- Leg: a = √(6² + 12²) = √180 ≈ 13.42 ft, the sloped length from wall to ridge before any overhang.
- Area: ½ × 24 × 6 = 72 ft² of triangular wall to side or paint.
- Base angles: arctan(6 ÷ 12) ≈ 26.57°, the slope of the roof.
- Apex angle: 180° − 2 × 26.57° ≈ 126.87°, so this is an obtuse isosceles triangle.
- Perimeter: 2 × 13.42 + 24 ≈ 50.83 ft of trim around the gable.
Which inputs fix the triangle
| You know | What the calculator finds first | Limit |
|---|---|---|
| Leg and base | h = √(a² − (b/2)²) | b < 2a |
| Base and height | a = √(h² + (b/2)²) | none |
| Leg and height | b = 2√(a² − h²) | h < a |
| Leg and apex angle | b = 2a sin(γ/2), h = a cos(γ/2) | 0° < γ < 180° |
| Base and base angle | h = (b/2) tan α, a = (b/2) ÷ cos α | α < 90° |
| Base and apex angle | h = (b/2) ÷ tan(γ/2), a = (b/2) ÷ sin(γ/2) | 0° < γ < 180° |
Two angles are never enough on their own: an isosceles triangle with 70° base angles can be the size of a postage stamp or a mountainside.
The base angles theorem
Euclid proved in Book I, Proposition 5 of the Elements that the angles at the base of an isosceles triangle are equal. Medieval students nicknamed the proof the pons asinorum, the “bridge of asses”, because it was the first real hurdle in the book. The converse, Proposition 6, says two equal angles force two equal sides. Together they explain why the apex altitude is also the median, angle bisector and perpendicular bisector of the base, which every formula above relies on.
Special isosceles triangles
- Right isosceles (45°-45°-90°): the apex angle is 90° and the base is a√2. With 10 cm legs the base is about 14.14 cm and the area is 50 cm². It is half of a square cut along its diagonal.
- Equilateral: when the base equals the leg, every angle is 60°. The equilateral triangle calculator handles that case from a single measurement.
- Golden triangle (36°-72°-72°): with 10 cm legs and a 36° apex, the base is about 6.18 cm, so leg ÷ base ≈ 1.618, the golden ratio. Each point of a regular five-pointed star is one of these.
- A-frames: a cabin with 24 ft rafters meeting at a 50° apex stands about 21.75 ft tall on a 20.29 ft wide footprint, with about 220.6 ft² of gable wall.
For triangles with no equal sides, use the general triangle calculator, or the right triangle calculator when one angle is 90°.
Frequently asked questions
Are the base angles of an isosceles triangle always equal?
Yes. The angles opposite the two equal sides are equal; this is the base angles theorem, Proposition 5 in Book I of Euclid's Elements. The converse also holds: a triangle with two equal angles has two equal sides.
How do I find the area of an isosceles triangle from the legs and base?
First find the height with h = √(a² − (b/2)²), then use A = ½ × b × h. With 5 cm legs and a 6 cm base, h = √(25 − 9) = 4 cm and the area is 12 cm².
How many measurements do I need?
Two independent ones, and at least one must be a length. A base angle already fixes the apex angle (γ = 180° − 2α), so angles alone give the shape but never the size.
Why does the calculator reject my numbers?
Some combinations cannot close into a triangle. The base must be shorter than twice the leg, the height must be shorter than the leg, each base angle must be under 90°, and the apex angle must be between 0° and 180°.
What is the difference between 'height to base' and 'height to a leg'?
Height to base is the usual altitude from the apex straight down to the base. Height to a leg runs from a base corner perpendicular to the opposite leg; it equals b × h ÷ a and matters when the triangle rests on one of its legs.