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Isosceles Triangle Calculator

Solve an isosceles triangle's area, height and perimeter at once.

Result

36.66
9.17
28.00

How it is calculated

You enter the base b and the length of the two equal sides a, in one unit. The height to the base is found with Pythagoras on half the triangle: h = √(a² − (b ÷ 2)²). The area is b × h ÷ 2 and the perimeter is 2a + b. Results are rounded to two decimals, and the area is in the square of your unit. The tool only returns a result when the base is above zero and the equal side is strictly longer than half the base, because otherwise the two sides cannot meet above the base. An equal side exactly half the base would make a flat, zero-height shape and is rejected. A base equal to the side is allowed and gives an equilateral triangle.

Worked example

A triangle has a base of 8 and two equal sides of 10. Half the base is 4. Height: √(10² − 4²) = √(100 − 16) = √84 = 9.165, shown as 9.17. Area: 8 × 9.165 ÷ 2 = 36.66 square units, computed from the unrounded height. Perimeter: 2 × 10 + 8 = 28.00. If you shorten the sides to 5, the half-base of 4 is almost the full side, and the height falls to 3.

Try this example →

Common mistakes

  • Typing the height instead of the equal side. The second box is the slanted side, which is always longer than the height.
  • Entering an equal side that is too short. If it is half the base or less, the sides cannot reach each other and the tool gives no result. A base of 10 needs a side longer than 5.
  • Entering the unequal side as the equal side. The base is the one side that differs, the other two must be the equal pair, so check which side is which.
  • Using the area formula base × side ÷ 2, which is correct only for a right triangle. Here the height must come from Pythagoras first.
  • Rounding the height before computing the area by hand. The tool uses the unrounded height, so a hand check with 9.17 gives 36.68 rather than 36.66.

About this calculator

An isosceles triangle has two equal sides (the “legs”) and a base of a different length, which gives it two useful shortcuts once you know the base b and equal side a. Its height, drawn from the apex to the midpoint of the base, follows directly from the Pythagorean theorem: h = √(a² − (b/2)²). The area then works out to A = (b/4)√(4a² − b²), and the perimeter is simply P = 2a + b.

The defining symmetry of an isosceles triangle — the altitude to the base also bisects the base and the apex angle — is what makes it a favorite in geometry proofs, but it’s just as common in the physical world: roof trusses and gable ends are frequently built as isosceles triangles for structural symmetry, and architects and engineers rely on the same height and area formulas to size roof pitches, A-frame structures, and triangular support brackets.

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