Isosceles Triangle Calculator

Enter the two equal leg lengths and the base of an isosceles triangle to get its area, height, perimeter, and base/apex angles.

Quick Facts

Height formula
h = √(a² − (b/2)²)
The altitude from the apex bisects the base at a right angle.
Area formula
A = ½ × b × h
Same base-times-height rule as any triangle.
Base angles
β = arccos((b/2) / a)
The two base angles are always equal in an isosceles triangle.

Your Results

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

Ready

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

Formula and Method for the Isosceles Triangle Calculator

An isosceles triangle has two sides of equal length, called the legs (a), and a third side called the base (b). Because two sides are equal, the two angles opposite them — the base angles — are also equal, and the angle between the two legs is the apex angle. This calculator takes the leg length and base length and derives the height, area, perimeter, and all three angles.

How the calculation works

Drop a perpendicular from the apex (where the two equal legs meet) straight down to the base. That altitude bisects the base into two equal halves of length b/2, and it splits the isosceles triangle into two congruent right triangles. Applying the Pythagorean theorem to one of those right triangles gives the height: h = √(a² − (b/2)²). From there, the area follows the usual triangle rule, A = ½ × b × h, and the perimeter is simply P = 2a + b. For the angles, the right triangle also gives cos(β) = (b/2)/a, so the base angle is β = arccos((b/2) / a), and since all three angles of a triangle sum to 180°, the apex angle is α = 180° − 2β.

Common mistakes

  • Confusing the leg with the base: the leg (a) is one of the two equal sides; the base (b) is the remaining, generally different, side. Swapping them changes every downstream result.
  • Ignoring the triangle inequality: a valid isosceles triangle requires 2a > b — the two legs together must be longer than the base — otherwise the shape cannot close and no real height exists (the value under the square root would be negative).
  • Mixing units: enter the leg and base in the same unit before calculating; area comes out in that unit squared (e.g., ft²), not the base unit.

Real-world applications

  • Roof trusses, A-frames, and gable ends are commonly isosceles triangles, where the height determines clearance and the angles determine the roof pitch.
  • Surveying and construction use the base-angle formula to lay out symmetric structures without measuring the apex angle directly.
  • Design and engineering drawings use the height formula to confirm a triangular bracket or gusset fits within a given clearance.
  • Geometry and trigonometry coursework use isosceles triangles as the standard example for the base-angles-are-equal theorem and for practicing the Pythagorean theorem on the altitude.

Frequently Asked Questions

What is an isosceles triangle?
An isosceles triangle has two sides (the legs) of equal length and a third side (the base) that may differ. Because two sides are equal, the two angles opposite them — the base angles — are also equal.
How do you find the area of an isosceles triangle from its legs and base?
First find the height from the apex to the base: h = √(a² − (b/2)²), where a is the leg length and b is the base. Then the area is A = ½ × b × h. For legs of 10 and a base of 12, h = √(100 − 36) = 8, so A = ½ × 12 × 8 = 48.
How do you find the apex angle and base angles?
The base angles (equal) are β = arccos((b/2) / a), and the apex angle is α = 180° − 2β. Using the law of cosines directly on the apex, α = arccos((2a² − b²) / (2a²)). All three angles always sum to 180°.
What values are valid for the legs and base?
Both the leg length a and base length b must be positive, and the triangle inequality requires 2a > b — the two legs together must be longer than the base, or no triangle can be formed.

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