Isosceles & Equilateral Triangle Calculator

Quickly calculate dimensions, area, perimeter, heights, and angles for isosceles and equilateral triangles.

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Side length (all three sides)

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Isosceles & equilateral triangle calculator

An equilateral triangle has all three sides equal — and therefore all three angles equal to 60°. An isosceles triangle has at least two equal sides; the two equal sides face equal base angles, and the third side is the base.

These shapes have closed-form expressions for their area, height, inradius, and circumradius that are simpler than the general triangle formulas. Toggle between modes and the calculator returns everything you need from a single input (equilateral: side length) or two inputs (isosceles: equal side + base).

How the closed-form solutions work

Equilateral — area

All vertices lie on the circle of radius a/√3 centred at the centroid. Drop one altitude and the half-base-altitude rectangle gives Area = (√3 / 4) · a².

Equilateral — height

By the Pythagorean theorem on the half-triangle, h = (√3 / 2) · a. The centroid, incenter, circumcenter, and orthocenter all coincide.

Isosceles — drop the altitude

Splitting the isosceles triangle along its axis of symmetry produces two congruent right triangles. Their height is h = √(a² − (b/2)²) where a is the equal side and b the base.

Base and apex angles

The base angles are equal: B = arccos((b/2) / a). The apex angle is 180° − 2B.

Ways to use this calculator

Roof gables and dormers

Equilateral or isosceles roof framing — given the span (base) and rafter length (equal side), find the rise (height).

Decoration and pattern design

Tile, quilt, and tessellation designs frequently rely on equilateral or isosceles units; solve once and reuse.

Antenna and lattice work

Equilateral truss panels are common because their symmetry distributes load evenly.

Geometry homework

Closed-form expressions make these shapes ideal for first proofs of Pythagoras and basic trig identities.

Verifying construction tolerances

Measure two sides and a base — if they match the closed-form predictions to within tolerance, your isosceles assembly is square.

Quick area estimates

When base and equal side are easy to measure, area follows from one square root and one multiplication.

Best practices

Two equal sides must each exceed half the base

Otherwise the triangle inequality fails and the figure collapses.

Use the equilateral mode for symmetry checks

If your isosceles triangle's equal side and base happen to match, switch to equilateral and verify all six measurements.

Equilateral & isosceles formulas

Equilateral area

Area = (√3 / 4) · a²

Equilateral height

h = (√3 / 2) · a

Equilateral inradius

r = a / (2√3)

Equilateral circumradius

R = a / √3

Isosceles height

h = √(a² − (b/2)²)

Isosceles area

Area = ½ · b · h

Isosceles base angle

B = arccos((b/2) / a)

Isosceles apex angle

Apex = 180° − 2B

Frequently asked questions

For an equilateral triangle of side a, the area is (√3 / 4) · a². For example, a side of 9 gives Area = (√3 / 4) · 81 ≈ 35.0741. This follows from dropping an altitude — it has length (√3 / 2) · a — and using ½ · base · height = ½ · a · (√3 / 2) · a.

The altitude of an equilateral triangle of side a is h = (√3 / 2) · a ≈ 0.866 · a. All three altitudes are the same length, and they all coincide with the three medians, perpendicular bisectors, and angle bisectors — the four classical centres collapse to a single point at the geometric centre.

Drop a perpendicular from the apex to the midpoint of the base. The Pythagorean theorem on the resulting right triangle gives h = √(a² − (b/2)²), where a is the equal side and b is the base. From the height you get the area directly: Area = ½ · b · h.

The two equal sides face two equal base angles B = arccos((b/2) / a). The third angle (the apex) is 180° − 2B. For an isosceles triangle with equal sides 10 and base 12, the base angles are arccos(0.6) ≈ 53.13° each and the apex is ≈ 73.74°.

Yes — every equilateral triangle is an isosceles triangle (in fact, all three pairs of sides are equal, so it satisfies the "at least two equal sides" definition of isosceles in every way). The reverse is not true: an isosceles triangle with unequal apex and base angles is not equilateral.

For side a, the inradius is r = a / (2√3) and the circumradius is R = a / √3. The ratio R / r = 2 — a fact unique to equilateral triangles. Both centres sit at the same point as the centroid and orthocenter, and that point lies one-third of the altitude above the base.