Last updated: 2026-05-18

Definition

An isosceles triangle has two equal sides (the legs, each of length $a$) and a third side (the base, length $b$). The two angles at the base are equal (base angles $\beta$); the angle at the apex between the two legs is the apex angle $\theta$. This symmetry means two independent measurements fully determine the triangle.

Input modes

The calculator accepts the triangle in either of two ways:

Mode 1 — Leg & Base: the two equal legs $a$ and the base $b$.
Mode 2 — Leg & Apex Angle: one leg $a$ and the apex angle $\theta$.

Both modes compute the height, area, perimeter, circumradius, and inradius.

Mode 1: From Leg and Base

Drop the altitude from the apex to the midpoint of the base. This splits the triangle into two congruent right triangles with hypotenuse $a$ and base leg $b/2$.

Height: $h = \sqrt{a^2 - \left(\frac{b}{2}\right)^2}$

Area: $S = \frac{b \cdot h}{2}$

Base angle: In the right triangle, $\cos\beta = \tfrac{b/2}{a} = \tfrac{b}{2a}$, so: $\beta = \arccos\!\frac{b}{2a}$

Apex angle: $\theta = 180° - 2\beta$

Worked example — $a = 5$, $b = 6$:

$h = \sqrt{25 - 9} = 4, \quad S = \frac{6 \times 4}{2} = 12$

$\beta = \arccos(0.6) \approx 53.13°, \quad \theta = 180° - 106.26° = 73.74°$

Mode 2: From Leg and Apex Angle

Given $a$ and $\theta$, split the apex angle in half. Each half-angle is $\theta/2$.

Base: $b = 2a\sin\!\frac{\theta}{2}$

Height: $h = a\cos\!\frac{\theta}{2}$

Area (SAS formula): $S = \frac{1}{2}\,a^2\sin\theta$

Base angle: $\beta = \frac{180° - \theta}{2}$

Worked example — $a = 5$, $\theta = 90°$:

$b = 2 \times 5 \times \sin 45° = 10 \times \tfrac{\sqrt{2}}{2} = 5\sqrt{2} \approx 7.071$

$h = 5 \times \cos 45° = \tfrac{5\sqrt{2}}{2} \approx 3.536, \quad S = \tfrac{1}{2} \times 25 \times \sin 90° = 12.5$

This is a 45-45-90 (right isosceles) triangle.

Circumradius and Inradius

Circumradius (using $R = abc \div (4S)$ with $c = b$, two sides $= a$):

$R = \frac{a^2 b}{4S}$

Inradius (using $r = S \div s$ where $s = (2a + b)/2$):

$r = \frac{S}{s}$

Special Cases

When $a = b$, the triangle is equilateral ($\theta = 60°$). When $b \geq 2a$, the triangle inequality fails and no triangle is possible.