What is an Isosceles Triangle?
An isosceles triangle is a triangle with two sides of equal length, called legs (a). The third side is called the base (b). The angle between the two equal legs is the vertex angle (α), and the two equal angles at the base are the base angles (β).
Key properties of isosceles triangles:
- It has an axis of symmetry along its vertex height (the altitude from the apex to the base midpoint).
- The two base angles are always equal: β = (180° − α) / 2.
- The isosceles triangle can be acute, right, or obtuse — it depends on the vertex angle.
- An equilateral triangle (all sides equal) is a special case of an isosceles triangle.
Isosceles Triangle Formulas for Area and Perimeter
Given leg a and base b, the most useful formulas are:
Height from Apex (h)
The altitude from the vertex angle to the base:
h = √(a² − b²/4) = (1/2) × √(4a² − b²)
Area (A)
A = (b × h) / 2 = (b/4) × √(4a² − b²)
You can also use: A = (1/2) × a × b × sin(β) = (1/2) × a² × sin(α)
Perimeter (P)
P = 2a + b
Inradius (r) — Inscribed Circle Radius
r = A / s, where s = P/2 = (2a + b)/2 is the semi-perimeter.
Circumradius (R) — Circumscribed Circle Radius
R = a² × b / (4A)
Angles
Vertex angle: α = 2 × arcsin(b / 2a)
Base angles: β = (180° − α) / 2 = arccos(b / 2a)
What is the Isosceles Triangle Theorem?
The isosceles triangle theorem (also called the base angles theorem) states:
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
The converse is also true: if two angles of a triangle are congruent, then the sides opposite those angles are congruent. This means you can determine if a triangle is isosceles from either its sides or its angles.
Golden Triangle
A special isosceles triangle is the golden triangle (also called the sublime triangle), where the ratio of leg to base equals the golden ratio: a / b = φ = (1 + √5) / 2 ≈ 1.618. This triangle has base angles of 72° and a vertex angle of 36°. It appears frequently in pentagons, the Parthenon, and other classical structures.
How to Use This Isosceles Triangle Calculator
- Select your unit system: Metric (mm, cm, m, km) or US Imperial (in, ft, yd, mi).
- Choose the specific unit from the dropdown.
- Enter the leg length (a) — the two equal sides.
- Enter the base length (b) — the unequal third side.
- Click Calculate to instantly get all triangle properties.
The calculator will check if your triangle is a golden triangle and show a special notice if it is.
FAQs
What makes a triangle isosceles?
A triangle is isosceles when exactly two of its three sides are equal in length. The equal sides are called legs, and the unequal side is called the base.
Can an isosceles triangle be a right triangle?
Yes! A 45-45-90 triangle is both isosceles and right-angled. The two legs are equal, and the vertex angle is 90°.
What is the constraint for an isosceles triangle?
For a valid isosceles triangle, the base must be strictly less than twice the leg: b < 2a. This follows from the triangle inequality.
How is the height of an isosceles triangle calculated?
The height from the apex to the base bisects the base. Using the Pythagorean theorem on the resulting right triangle: h = √(a² − (b/2)²).
What are the base angles of an isosceles triangle?
The two angles at the base are always equal. If the vertex angle is α, then each base angle β = (180° − α) / 2.