Isosceles Triangle Calculator
Enter the length of the two equal sides and the base to find the area, perimeter, and height of your isosceles triangle. Simple, fast, and accurate.
Geometry of Isosceles Triangles
Isosceles triangles have a line of symmetry running from the vertex angle (the angle between the two equal sides) down to the midpoint of the base. That symmetry means the height, median, angle bisector, and perpendicular bisector from the vertex angle are all the same line.
The height divides the isosceles triangle into two congruent right triangles. Each right triangle has hypotenuse equal to the equal side a, one leg equal to half the base (b/2), and the other leg equal to the height h. The Pythagorean theorem gives h² + (b/2)² = a², so h = √(a² - b²/4).
Once you have the height, the area follows from A = ½ × base × height. The perimeter is simply the sum of all three sides: P = 2a + b. These formulas make isosceles triangle calculations straightforward as long as you know the equal side and the base.
Real-World Uses for Isosceles Triangles
Isosceles triangles show up in architecture and design whenever symmetry is desired. Roof trusses often use isosceles triangles to distribute weight evenly. Church spires, A-frame cabins, and decorative arches rely on isosceles geometry for both aesthetic appeal and structural stability.
In navigation, isosceles triangles help with triangulation. If you know the distance to two landmarks and the angle between them, you can model your position using an isosceles triangle. Surveyors use similar principles when measuring land boundaries.
Even everyday objects like coat hangers, ladder rungs, and certain kites approximate isosceles shapes. Understanding how to calculate their properties helps in design, manufacturing, and quality control.
Isosceles vs. Equilateral vs. Scalene
An equilateral triangle is a special isosceles triangle where the base also equals the two equal sides. In that case, all three sides are identical, and all three angles are 60 degrees. This calculator handles the general isosceles case where the base can differ.
Scalene triangles have all three sides different. They lack the symmetry of isosceles triangles, so you cannot assume the height bisects the base or that any two angles are equal. Scalene triangles require more information (all three sides, or two sides and an angle, or other combinations) to calculate area and other properties.
This calculator is designed specifically for isosceles triangles. If your triangle has three different side lengths, use a general triangle calculator or Heron's formula instead.
Frequently Asked Questions
What is an isosceles triangle?
An isosceles triangle has two sides of equal length (the legs) and a third side called the base. The two angles opposite the equal sides are also equal.
How do you find the height of an isosceles triangle?
Drop a perpendicular from the vertex between the equal sides to the base. It bisects the base, creating two right triangles. Use the Pythagorean theorem: h = √(a² - (b/2)²), where a is the equal side and b is the base.
What is the area formula?
Once you have the height h and base b, the area is A = (b × h) / 2. This calculator finds the height automatically from the equal side and base.
Can an isosceles triangle be a right triangle?
Yes. If the two equal sides are the legs and they meet at a 90-degree angle, you have an isosceles right triangle with base angles of 45 degrees each.
What happens if the base is too long?
The two equal sides must satisfy the triangle inequality: 2a > b. If the base equals or exceeds twice the equal side length, the triangle cannot exist.