Isosceles Triangle Calculator

Background.

Fold a paper isosceles triangle so its two equal sides lie exactly on top of each other. The crease runs from the top corner straight down to the middle of the bottom edge and meets it at a right angle. Nearly every calculation for this shape is a consequence of that fold.

The crease is the height. It splits the triangle into two identical right triangles, each with one of the equal sides as its hypotenuse and half the bottom edge as its short side, so Pythagoras gives the height directly: √(leg² − (base ÷ 2)²). Legs of 13 over a base of 10 stand exactly 12 tall, because 13² − 5² = 144. The angles come from the same half-triangle, and the area is half the base times that height.

Because the shape has only two degrees of freedom, any two measurements settle it. You can start from a leg and the base, the base and the height, a leg and the height, or either side length together with the apex angle, and get back every other length, both angles, the area, the perimeter and the radii of the inscribed and circumscribed circles.

Most wrong answers come from mixing up the names. The legs are the two equal sides. The base is the odd one out, and it can be longer or shorter than a leg. The apex angle sits at the top where the legs meet; the two base angles sit at the bottom corners and are always equal to each other.

What is isosceles triangle calculator?

An isosceles triangle has at least two sides of equal length, called the legs; the third side is the base, the angle between the legs is the apex angle, and the two angles standing on the base are the base angles. Euclid proved in Book I, Proposition 5 of the Elements — the pons asinorum, or bridge of asses — that the base angles are equal, and Proposition 6 proves the converse: two equal angles force two equal sides. So apex + 2 × base angle = 180°, and one angle determines the other two. The line from the apex perpendicular to the base is the axis of symmetry, and in this shape it is simultaneously the altitude, the median to the base, the perpendicular bisector of the base and the bisector of the apex angle.

How to use this calculator.

1. 01Choose the pair of measurements you actually have under 'Which two measurements do you know?'. Leg and base is the usual textbook case; base and height suits a gable or a tent where you know the width and the rise.
2. 02Fill in the two fields for that pair and ignore the others. Every length must be in the same unit, and the area comes back in that unit squared.
3. 03If your angle is one of the bottom corners rather than the top, convert it first: apex angle = 180° − 2 × base angle. A base angle of 70° means an apex angle of 40°.
4. 04Respect the two limits: the base must be shorter than twice the leg, and in leg-and-height mode the height must be shorter than the leg. Either way round, the legs could not meet above the base.
5. 05Read the height first, then the angles. The base-angle output is a single figure because both base angles are the same.

The formula.

Whatever pair you enter is first turned into a leg and a base. From base and height, leg = √(h² + (base ÷ 2)²). From leg and height, base = 2√(leg² − h²). From a leg and the apex angle θ, base = 2 × leg × sin(θ ÷ 2); from the base and θ, leg = base ÷ (2 sin(θ ÷ 2)). The last two come from the half-triangle, where half the base is opposite half the apex angle and the leg is the hypotenuse.

With leg and base fixed, everything else follows. Height h = √(leg² − (base ÷ 2)²), which is Equation (1) in MathWorld's isosceles-triangle entry; note that MathWorld letters the base a and the legs b, the reverse of the names used here. Area A = ½ × base × h. Perimeter P = 2 × leg + base. The apex angle is 2 × arcsin((base ÷ 2) ÷ leg), and each base angle is arccos of the same ratio, (base ÷ 2) ÷ leg.

The two radii use the relations that hold for every triangle: the inradius is r = A ÷ s, where s is half the perimeter, and the circumradius is R = leg × leg × base ÷ (4A), the familiar abc ÷ 4K with two of the three sides equal. MathWorld's isosceles-only versions, r = a(√(a² + 4h²) − a) ÷ 4h and R = (a² + 4h²) ÷ 8h with a as the base, give identical values. When the apex angle exceeds 90° the circumcentre lies outside the triangle, below the base.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Wolfram Research, MathWorld — ‘Isosceles Triangle’ (live revision retrieved 2026-07-29). Equation (1) h = √(b² − ¼a²); the area form ½ah; Equation (7) inradius r = a(√(a²+4h²) − a)/(4h); Equation (8) circumradius R = (a²+4h²)/(8h). Note that the entry's lettering is the reverse of this page's — its a is the base and its b is the legs — so its Equation (1) reads √(leg² − ¼·base²) here. The entry gives no closed form for the base angles; those are derived on this page from the right-half decomposition. Open access.
2. [2]Euclid, Elements, Book I, Proposition 5 — ‘In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.’ The pons asinorum, and the reason a single apex angle determines the whole shape. David E. Joyce web edition of the Heath translation, Clark University; quotation confirmed on retrieval 2026-07-29.
3. [3]Euclid, Elements, Book I, Proposition 47 (the Pythagorean theorem) — the theorem behind height = √(leg² − (base/2)²) once the symmetry axis splits the triangle into two congruent right halves. Joyce web edition, Clark University; retrieved 2026-07-29.
4. [4]Wolfram Research, MathWorld — ‘Triangle’ (live revision retrieved 2026-07-29), Equation (18) Δ = ½ah, the general base-and-height area form used here with the base and its altitude. Open access.