Isosceles Triangle Side Calculator

Enter the base and height of an isosceles triangle to find the length of its two equal sides (legs) using the Pythagorean theorem, plus its perimeter, area, and base angles.

Quick Facts

Leg formula
a = √((b/2)² + h²)
The Pythagorean theorem applied to the right triangle formed by the height, half the base, and the leg.
Perimeter formula
P = b + 2a
Base plus the two equal legs.
Area formula
A = ½ × b × h
Base times height, divided by two — the same formula as any triangle.
Base angle formula
β = arctan(h ÷ (b/2))
The two base angles of an isosceles triangle are always equal.

Your Results

Calculated
Leg Length (a)
-
a = √((b/2)² + h²)
Perimeter
-
P = b + 2a
Area
-
A = ½ × b × h
Base Angles
-
β = arctan(h ÷ (b/2))

Ready

Enter the base and height, then press Calculate.

Formula and Method for the Isosceles Triangle Side Calculator

An isosceles triangle has two sides of equal length (the legs) and a third side (the base) that is usually a different length. Because it is symmetric, the perpendicular line from the apex (the vertex between the two equal legs) down to the base always bisects the base and meets it at a right angle. That single fact is what lets you find the leg length from just the base and the height.

How the calculation works

Dropping the altitude from the apex to the base splits the isosceles triangle into two congruent right triangles. Each right triangle has one leg equal to half the base (b/2), the other leg equal to the height (h), and a hypotenuse equal to the isosceles triangle's equal side (a). By the Pythagorean theorem: a = √((b/2)² + h²). Once you know the leg, the perimeter follows directly as P = b + 2a, and the area uses the standard triangle formula A = ½ × b × h (it doesn't need the leg length at all). The base angles — equal by definition in an isosceles triangle — come from the same right triangle: β = arctan(h ÷ (b/2)), and the apex angle is 180° − 2β.

Common mistakes

  • Using the full base instead of half: the Pythagorean relationship uses b/2, not b — forgetting to halve the base roughly doubles the computed leg length.
  • Confusing height with leg length: the height is the perpendicular distance from the apex straight down to the base, not the length of a slanted side.
  • Mixing units: enter the base and height in the same unit — converting one but not the other produces a leg length that is neither right nor easy to spot as wrong.

Real-world applications

  • Roof trusses and gable ends are frequently isosceles, so builders use this relationship to cut rafters of the correct length from the span and rise.
  • A-frame structures, tents, and easels rely on the same base-height-to-leg relationship to determine strut or leg length.
  • Sign, kite, and pennant fabrication uses the formula to cut symmetric fabric or board panels to size.
  • Surveying and CAD work use it to verify that a drawn or measured triangle is truly isosceles before relying on its symmetry elsewhere in a design.

Frequently Asked Questions

How do you find the side (leg) length of an isosceles triangle from its base and height?
Drop a perpendicular from the apex to the base; it splits the isosceles triangle into two congruent right triangles with legs of b/2 and h. By the Pythagorean theorem, the equal side (leg) is a = √((b/2)² + h²). For example, a triangle with an 8 ft base and 5 ft height has legs of √(4² + 5²) = √41 ≈ 6.40 ft.
What is the relationship between the leg, base, and height of an isosceles triangle?
The height h, half the base (b/2), and the leg a form a right triangle, so a² = (b/2)² + h². Rearranged, this also gives h = √(a² − (b/2)²) if you know the leg and base instead of the height.
How do I find the base angles of an isosceles triangle?
The two base angles are equal and can be found with β = arctan(h ÷ (b/2)). The apex angle is then 180° minus twice the base angle.
How is the area of an isosceles triangle calculated?
Area uses the same formula as any triangle: A = ½ × base × height. You don't need the leg length to calculate area, only the base and the perpendicular height.

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