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Isosceles triangle solver

With equal sides of 5 and a base of 6, the isosceles triangle solver finds a height of 4, an area of 12 and base angles of 53.13°.

degrees

degreesFor ASA, AAS and SSA. In ASA the side between angles A and B is c; in AAS the side given is a; in SSA the sides are a and b.

degreesFor ASA and AAS.

Area12

Area Heron's formula from the three sides: s = 8, area = sqrt(s(s-a)(s-b)(s-c)).
12
Sides
a = 5, b = 5, c = 6
Angles Summing to 180 degrees, as every plane triangle must.
A = 53.13deg, B = 53.13deg, C = 73.74deg
Area
12
Perimeter
16
Type
isosceles, acute
Heights The perpendicular from each side to the opposite vertex. Area is half the base times the height for any of the three.
to a: 4.8, to b: 4.8, to c: 4
Circumradius The radius of the circle through all three vertices.
3.125
Inradius The radius of the largest circle that fits inside.
1.5
Common Pythagorean triples
abcMultiples
3456-8-10, 9-12-15, 30-40-50
5121310-24-26, 15-36-39
8151716-30-34
7242514-48-50
20212940-42-58
9404118-80-82
  • The Pythagorean theorem holds only for right triangles. For any other triangle the law of cosines generalizes it: c squared equals a squared plus b squared minus 2ab cos(C), which reduces to Pythagoras when C is 90 degrees and cos(C) is zero.
  • Any two sides must add to more than the third, or the shape does not close. That triangle inequality is checked here before anything else is computed.

How the Triangle Calculator works

An isosceles triangle has two equal sides, the legs, and a third side, the base. The angles opposite the legs, the base angles, are equal too. With legs a and a base b, the height to the base is h = √(a² - (b ÷ 2)²), and the area is ½ × b × h.

For legs of 5 and a base of 6, half the base is 3, so h = √(25 - 9) = 4, and the area is ½ × 6 × 4 = 12. The perimeter is 2 × 5 + 6 = 16.

The height cuts the triangle into two right triangles with sides 3, 4 and 5, so each base angle is arccos(3 ÷ 5) = 53.13°, and the apex angle is 180° - 2 × 53.13° = 73.74°.

The legs must be longer than half the base, or they cannot meet above it: legs of 3 on a base of 6 lie flat, and anything shorter does not close.

From the apex angle instead, with legs a and an apex angle θ, the base is 2a sin(θ ÷ 2) and the area is ½a² sin θ. For legs of 5 and an apex of 73.74°, that is a base of 6 and an area of 12 again.

Triangle Calculator FAQ

What is the formula for the area of an isosceles triangle?

A = ½ × b × √(a² - b² ÷ 4), where a is a leg and b the base. Legs of 10 on a base of 12 have a height of √(100 - 36) = 8, so the area is ½ × 12 × 8 = 48.

What is an isosceles right triangle?

One whose apex angle is 90°, so each base angle is 45° and the base is a leg times √2. With legs of 1, the base is 1.4142.

How do I find the base angles from the apex angle?

Subtract the apex angle from 180° and halve what is left. An apex of 40° gives base angles of 70° each.

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