An isosceles triangle calculator finds the area, perimeter, height and angles of a triangle that has two equal sides. Choose what you know: the base and the equal side, the base and the height, or the equal side and the apex angle. The calculator fills in everything else and shows the steps.
If you only know the three side lengths, the Heron’s formula calculator gives the area for any triangle.
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How the isosceles triangle calculator works
Parts of an isosceles triangle
The two equal sides are the legs and the third side is the base. The angle between the legs is the apex angle, and the two angles at the base are equal. The height is the line from the apex straight down to the middle of the base.
Formulas
Drawing the height splits the triangle into two right triangles, each with half the base as one side.
Height = √(Leg² − (Base ÷ 2)²)
Area = Base × Height ÷ 2
Perimeter = Base + 2 × Leg
Base angle = arctan(Height ÷ (Base ÷ 2))
Apex angle = 180° − 2 × Base angle
When you know the leg and the apex angle
Base = 2 × Leg × sin(Apex angle ÷ 2)
Height = Leg × cos(Apex angle ÷ 2)
When you know the base and the height
Leg = √((Base ÷ 2)² + Height²)
Worked examples
Example 1: Base 6 and leg 5
Height = √(5² − 3²) = √16 = 4
Area = 6 × 4 ÷ 2 = 12
Perimeter = 6 + 2 × 5 = 16
Each base angle is arctan(4 ÷ 3) ≈ 53.13 degrees, so the apex angle is about 73.74 degrees.
Example 2: Base 10 and height 12
Leg = √(5² + 12²) = √169 = 13
Area = 10 × 12 ÷ 2 = 60
Perimeter = 10 + 2 × 13 = 36
The base angles are about 67.38 degrees each and the apex angle is about 45.24 degrees.
Example 3: Leg 10 and apex angle 40 degrees
Base = 2 × 10 × sin(20°) ≈ 6.8404
Height = 10 × cos(20°) ≈ 9.3969
Area = 6.8404 × 9.3969 ÷ 2 ≈ 32.139
The base angles are (180 − 40) ÷ 2 = 70 degrees each, and the perimeter is about 26.84.
Checking that a triangle exists
The leg must be longer than half the base. If it is not, the two legs cannot meet above the base and no triangle exists. The calculator warns you when this happens.
Frequently asked questions
How do you find the area of an isosceles triangle?
Find the height with the Pythagorean theorem, then use area = base × height ÷ 2. With base 6 and leg 5 the height is 4, so the area is 12.
Are the base angles of an isosceles triangle equal?
Yes. The two angles opposite the equal sides are equal, and together with the apex angle they add up to 180 degrees.
How do I find the angles of an isosceles triangle?
Each base angle is arctan(height ÷ half the base). The apex angle is 180 degrees minus twice a base angle.
What if I only know the three sides?
Use Heron’s formula, which gives the area of any triangle from its sides. The two equal sides in an isosceles triangle are just a special case.
What units does the calculator use?
It uses whatever unit you enter for lengths. The area is in that unit squared, and the angles are in degrees.