Isosceles Triangle Calculator

Background.

The Quanta isosceles triangle calculator solves a two-equal-sided triangle from any two measurements: a leg and the base, the base and the height, a leg and the height, a leg and the apex angle, or the base and the apex angle. It returns the leg, the base, the height onto the base, the area, the perimeter, the apex angle, the two equal base angles, the inradius and the circumradius. With the default legs of 13 and base of 10 it returns a height of exactly 12, an area of exactly 60, a perimeter of 36, an apex angle of 45.2397298961°, base angles of 67.380135052° each, an inradius of 3.3333333333 and a circumradius of 7.0416666667.

Two measurements are enough because the shape has only two degrees of freedom. Once you fix the two equal sides and the odd one out, everything else follows — and Euclid explained why in the first book of the Elements. Proposition 5, the proposition medieval students called the pons asinorum or "bridge of asses", proves that the two angles at the base of an isosceles triangle are equal. That single fact is what lets a solitary apex angle stand in for two pieces of information, and it is why this page can accept an angle in place of a second length.

The vocabulary matters here more than on most geometry pages, because the two words are easy to swap. A **leg** is one of the two equal sides, running from the apex down to an end of the base. The **base** is the odd side out. The **apex angle** sits at the top between the two legs; the **base angles** sit at the two bottom corners and are equal to each other. The height reported here is the altitude onto the base, which in this shape is a single line doing four jobs at once: it is the altitude, the median to the base, the perpendicular bisector of the base and the bisector of the apex angle. It is not the altitude onto a leg, which is a different and generally less useful number.

That symmetry is also where all the arithmetic comes from. Drop the height from the apex and it lands exactly at the midpoint of the base, splitting the triangle into two congruent right triangles whose hypotenuse is a leg and whose short leg is half the base. The Pythagorean theorem then gives the height as √(leg² − (base ÷ 2)²), which is MathWorld's Equation 1 for the shape, and the angles come from the same right half: half the apex angle is arcsin((base ÷ 2) ÷ leg), and a base angle is arccos of the same ratio.

Two things about the input ranges are worth knowing before you type. The base is allowed to be longer than a leg — a wide, shallow isosceles triangle is perfectly ordinary, and it simply means the apex angle is obtuse and the circumcircle's centre falls outside the triangle. What the base cannot be is twice a leg or more: at exactly twice, the two legs lie flat along the base and the three corners are collinear, enclosing no area at all, so the calculator rejects that input rather than returning a height of zero. The mirror-image rule applies in leg-and-height mode: the height must be strictly less than the leg, because the leg is the hypotenuse of each half.

The equilateral triangle is the special case where the leg equals the base, and entering 10 for both here reproduces the equilateral page's numbers exactly — height 8.6602540378, area 43.3012701892, all angles 60°. In the other direction, an apex angle of 90° gives the right isosceles triangle whose base is leg × √2 and whose base angles are 45° each, the shape behind every set square and every 45-45-90 problem. For a triangle with three different sides, use the triangle solver or the triangle height page instead.

What is isosceles triangle calculator?

An isosceles triangle is a triangle with two sides of equal length. Those two are the legs; the third is the base; the angle between the legs is the apex angle and the two angles resting on the base are the base angles. Euclid, Elements Book I, Proposition 5 — the pons asinorum — establishes the shape's defining consequence: "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." Because those two angles are equal, the apex angle and each base angle determine one another through apex + 2 × base angle = 180°, which is why one angle plus one length pins the whole triangle down. Wolfram MathWorld's Isosceles Triangle entry gives the closed forms, though with the opposite lettering to this page — its a is the base and its b is the legs. Its Equation (1) is the height, h = √(b² − ¼a²), which reads √(leg² − ¼ × base²) in the words used here; the area is ½ah; Equation (7) gives the inradius as a(√(a² + 4h²) − a) ÷ (4h); and Equation (8) gives the circumradius as (a² + 4h²) ÷ (8h). This calculator uses the general relations instead — inradius = area ÷ semiperimeter and circumradius = leg × leg × base ÷ (4 × area) — and cross-checks both against MathWorld's isosceles-specific versions, which agree to ten decimal places. The line from the apex perpendicular to the base is the triangle's axis of symmetry, and in this shape alone it coincides with 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 have. Any two of leg, base, height and apex angle will do — the shape has only two degrees of freedom.
2. 02Check the vocabulary before typing: the leg is one of the two equal sides, the base is the odd one out, and the apex angle is at the top between the legs, not at the bottom corners.
3. 03Leg and base is the most common case, and the one where the height is the answer you want. The base must be less than twice the leg.
4. 04Base and height is the reverse: you know how wide and how deep, and you want the slanted sides, the angles and the perimeter.
5. 05Leg and apex angle suits a compass-and-protractor construction or a design specified by an angle — a roof pitch, a fan of equal spokes, a triangular sail.
6. 06Read the height at the top, then the base angles below. Both base angles are the same number, so the calculator reports it once rather than twice.
7. 07Use the same unit for every length. The area comes back in that unit squared; angles are always in degrees.

The formula.

Every mode reduces to a leg and a base, and everything else is then derived from those two. In leg-and-base mode the pair is given. In base-and-height mode, leg = √(height² + (base ÷ 2)²). In leg-and-height mode, base = 2√(leg² − height²). In leg-and-apex-angle mode, base = 2 × leg × sin(apex ÷ 2). In base-and-apex-angle mode, leg = base ÷ (2 × sin(apex ÷ 2)).

Once the leg and base are known, the height is √(leg² − (base ÷ 2)²), MathWorld's Equation 1 for the isosceles triangle. It comes straight from the symmetry: the altitude from the apex meets the base at its midpoint, so each half is a right triangle with hypotenuse equal to the leg and short leg equal to half the base, and Euclid I.47 does the rest. The area is ½ × base × height, the perimeter is 2 × leg + base, the apex angle is 2 × arcsin((base ÷ 2) ÷ leg) and each base angle is arccos of the same ratio.

Working the default all the way through, with legs of 13 and a base of 10: the height is √(13² − 5²) = √(169 − 25) = √144 = 12 exactly. The area is ½ × 10 × 12 = 60 exactly. The perimeter is 2 × 13 + 10 = 36. Half the apex angle is arcsin(5 ÷ 13) = arcsin(0.3846153846) = 22.619864948°, so the apex angle is 45.2397298961°. Each base angle is arccos(0.3846153846) = 67.380135052°, and the three add up: 45.2397298961 + 2 × 67.380135052 = 180 exactly. The semiperimeter is 18, so the inradius is 60 ÷ 18 = 3.3333333333, and the circumradius is 13 × 13 × 10 ÷ (4 × 60) = 1690 ÷ 240 = 7.0416666667.

Those last two are worth cross-checking, because this calculator deliberately uses the general triangle formulas rather than the isosceles-specific ones, so the two routes are an independent test of each other. MathWorld's Equation 7 for the inradius, in its own notation where a is the base, gives 10 × (√(100 + 4 × 144) − 10) ÷ (4 × 12) = 10 × (26 − 10) ÷ 48 = 3.3333333333. Its Equation 8 for the circumradius gives (100 + 4 × 144) ÷ (8 × 12) = 676 ÷ 96 = 7.0416666667. A third route, the extended law of sines, gives R = base ÷ (2 sin apex) = 10 ÷ (2 × 120 ÷ 169) = 7.0416666667. All three agree, and the test suite asserts all three. A fourth independent check on the area: Heron's formula on the sides 13, 13 and 10 gives a semiperimeter of 18 and an area of √(18 × 5 × 5 × 8) = √3600 = 60, exactly matching ½ × base × height.

Rounding stage: all arithmetic runs in Decimal at 40 significant digits and is rounded once, at the return boundary, to 10 decimal places. The leg and base are resolved at full precision and are never rounded before the height, area, angles and radii are derived from them. Decimal.js has no trigonometric functions, so arcsine, arccosine and sine are each evaluated once in double precision and immediately carried back into Decimal — an error floor near 1e-16. The three side-only modes reach the height through an exact square root with no trigonometry at all, which is why the worked example returns 12, 60 and 36 exactly. Nothing on this page is sorted into bands or grades, so no threshold depends on display rounding; the two validity edges (base < 2 × leg, height < leg) are input guards, and both are tested immediately below, at, and immediately above the boundary.

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.