Isosceles Triangle Area Calculator

Enter the two equal leg lengths and the base to compute the area, height, perimeter, and base/apex angles using A = ½ × base × height.

Quick Facts

Area formula
A = ¼b√(4a²−b²)
Equivalent to A = ½ × base × height once the height is found.
Height formula
h = √(a² − (b/2)²)
From the Pythagorean theorem on the leg, half-base, and altitude.
Validity condition
2a > b
Twice the leg length must exceed the base or no triangle exists.

Your Results

Calculated
Area
-
A = ½ × base × height
Height
-
h = √(a² − (b/2)²)
Perimeter
-
P = 2a + b
Base & Apex Angles
-
Base angles are equal; apex = 180° − 2×base

Ready

Enter the leg length, base length, and unit, then press Calculate.

Formula and Method for Isosceles Triangle Area

Area of an Isosceles Triangle:

A = ½ × base × height, where h = √(a² − (b/2)²)

combined: A = (b/4) × √(4a² − b²), where a is a leg (equal side) and b is the base

An isosceles triangle has two sides of equal length (the legs, a) and a third side (the base, b) that is typically a different length. Because the two legs are equal, the perpendicular dropped from the apex to the base always lands exactly at the base's midpoint, splitting the triangle into two congruent right triangles. That symmetry is what makes the area, height, and angles solvable directly from just a and b, without needing the general-purpose Heron's formula.

How the calculation works

The perpendicular from the apex to the base has length h = √(a² − (b/2)²) — a direct application of the Pythagorean theorem to the right triangle formed by the leg (hypotenuse), half the base, and the height. Once the height is known, the standard triangle area formula applies: A = ½ × base × height = ½ × b × √(a² − (b/2)²), which simplifies to A = (b/4) × √(4a² − b²). The perimeter is simply the sum of all three sides: P = 2a + b. The two base angles are equal (a defining property of isosceles triangles) and each equals arccos((b/2)/a); the apex angle is 180° minus twice that base angle.

Common mistakes

  • Confusing leg and base: the leg (a) is one of the two equal sides; the base (b) is the third, unequal side. Swapping them changes the shape entirely.
  • Invalid triangles: the triangle inequality requires 2a > b. If the base is too long relative to the legs (2a ≤ b), the two legs cannot meet above the base and no triangle exists.
  • Units: area comes out in square units (ft², m²) while height and perimeter stay in linear units (ft, m). Keep the leg and base in the same unit before entering them.

Real-world applications

  • Roof trusses, gables, and A-frame structures are commonly modeled as isosceles triangles when sizing material and pitch.
  • Cross-sections of dams, embankments, and road cuts often approximate isosceles triangles for area and volume estimates.
  • Signage, pennants, and decorative panels use the area to estimate material and the angles to plan cuts.
  • Geometry and trigonometry coursework use isosceles triangles to introduce the Pythagorean theorem and angle-side relationships.

Frequently Asked Questions

What is the formula for the area of an isosceles triangle?
Given the base b and the two equal leg lengths a, the area is A = (b/4) × √(4a² − b²). This comes from first finding the height h = √(a² − (b/2)²) and then applying the standard triangle area formula A = ½ × base × height.
How do you find the height of an isosceles triangle from its sides?
Drop a perpendicular from the apex to the base — it bisects the base and forms a right triangle with the leg as hypotenuse. By the Pythagorean theorem, height h = √(a² − (b/2)²), where a is a leg length and b is the base.
What values are valid for the base and legs?
The legs and base must be positive, and the triangle inequality must hold: twice the leg length must exceed the base (2a > b). If 2a ≤ b, the two legs cannot meet above the base and no triangle exists.
How do I find the base angles of an isosceles triangle?
The two base angles are equal because the triangle is isosceles. Each equals arccos((b/2)/a), where a is the leg length and b is the base. The apex angle is 180° minus twice the base angle.

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