Area of an Isosceles Triangle Calculator

Enter the equal side length and base — get area, height, perimeter, and step-by-step working instantly.

Unit:
Equal Side (a)
cm
Base (b)
cm

💡 Fill in both values — results appear instantly. Requires b < 2a.

Isosceles Triangle Diagram

Isosceles Triangle Formulas

Height

h = √(a² − (b/2)²)

Drop a perpendicular from apex to base midpoint (Pythagoras).

Area

A = (b × h) ÷ 2

Half the base times the height — standard triangle area formula.

Perimeter

P = 2a + b

Two equal legs plus the base.

Validity Check

b < 2a

The base must be less than 2× the equal side for a valid triangle.

What Is an Isosceles Triangle?

An isosceles triangle is a triangle with exactly two sides of equal length, called the legs. The third side is called the base. The two angles at the base (base angles) are also equal to each other — a consequence of the equal sides.

The line of symmetry of an isosceles triangle passes through the apex (top vertex) and the midpoint of the base. The height drawn from the apex to the base is both the altitude and the perpendicular bisector of the base, which is why we can use the Pythagorean theorem to find it.

a = a
Two equal legs of length a
b < 2a
Triangle inequality constraint
1 axis
One line of symmetry (unless equilateral)

How to Calculate the Area — Step by Step

  1. 1

    Know the equal side (a) and base (b)

    These are the two measurements needed. Confirm the triangle is valid: base b must be less than 2 × equal side a.

  2. 2

    Calculate the height using Pythagoras

    The height splits the isosceles triangle into two identical right triangles. Each has hypotenuse = a and base = b/2. So: h = √(a² − (b/2)²).

  3. 3

    Apply the area formula

    Area = ½ × base × height = ½ × b × h. The result is in square units (cm², m², etc.).

  4. 4

    Find the perimeter

    P = 2a + b — simply sum the two equal sides and the base.

Worked Example — a = 10 cm, b = 12 cm

Step 1: h = √(10² − 6²) = √(100 − 36) = √64 = 8 cm
Step 2: A = ½ × 12 × 8 = 48 cm²
Step 3: P = 2(10) + 12 = 32 cm

Common Isosceles Triangle Measurements

Equal Side a Base b Height h Area Perimeter
10 cm 12 cm 8.0 cm 48.0 cm² 32 cm
13 cm 10 cm 12.0 cm 60.0 cm² 36 cm
5 cm 6 cm 4.0 cm 12.0 cm² 16 cm
15 m 18 m 12.0 m 108.0 m² 48 m
7 cm 8 cm 5.74 cm 22.96 cm² 22 cm

Real-World Applications

🏛️

Architecture — Pediments

The triangular pediment on classical buildings (Greek temples, Roman columns) is typically an isosceles triangle for aesthetic symmetry.

⛺

Camping — Tent Shapes

Symmetrical tent fronts and A-frame structures are isosceles triangles, combining equal sides for structural stability and visual balance.

🚦

Signage — Warning Signs

Warning road signs, yield signs, and arrowhead shapes use isosceles triangles because they draw attention through symmetry.

🎨

Design — Decorative Elements

Symmetrical decorative triangles appear in logos, patterns, and geometric art, requiring accurate area calculations for material estimation.

📐

Education — Key Geometry Topic

Isosceles triangles are a cornerstone of school geometry in classes 7–10, introducing students to the Pythagorean theorem, symmetry, and base angle theorems.

Frequently Asked Questions

What is an isosceles triangle?
A triangle with exactly two equal sides (called the "legs") and a base. The two base angles are also equal. The word "isosceles" comes from Greek: isos (equal) + skelos (leg).
What is the formula for area of an isosceles triangle?
A = ½ × b × h where b is the base and h = √(a² − (b/2)²) is the height derived from the equal side a and half the base. For example, with a = 10 cm and b = 12 cm: h = √(100 − 36) = 8 cm → A = ½ × 12 × 8 = 48 cm².
How do you find the height of an isosceles triangle?
Drop a perpendicular from the apex to the midpoint of the base. This creates two identical right triangles, each with hypotenuse = a (equal side) and one leg = b/2 (half the base). By the Pythagorean theorem: h = √(a² − (b/2)²).
When is an isosceles triangle invalid?
The triangle is invalid when the base b ≥ 2a. This violates the triangle inequality — the sum of any two sides must be greater than the third side. Since both equal sides together = 2a, they must exceed the base. A b = 2a means the triangle degenerates to a flat line.
Is an equilateral triangle a special case of isosceles?
Yes — an equilateral triangle has all 3 sides equal, so any pair of sides qualifies as the "equal sides." It is a special isosceles triangle where the base also equals the equal sides (b = a), giving all three 60° angles. Use our Equilateral Triangle Calculator for that specific case.

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