Education

Isosceles Triangle Calculator

Calculate the main measurements of an isosceles triangle from the equal side length and base. The result includes height, area, perimeter, vertex angle, and equal base angles.

isosceles-triangle
Isosceles triangle
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What makes an isosceles triangle different?

An isosceles triangle has two equal sides. Because of that, the two base angles are also equal. OpenStax discusses triangle types and their properties in its section on triangles, side relationships, and angle relationships.

This calculator uses the common classroom setup: two equal sides and a base. From those three side lengths, it finds the height, area, perimeter, vertex angle, and base angles. That gives students enough information to answer most isosceles-triangle homework questions without jumping between multiple tools.

Formulas used

Equal sides = a Base = b Height = √(a² - (b/2)²) Area = (b × height) / 2 Perimeter = 2a + b

The height splits the isosceles triangle into two congruent right triangles. That is why the formula uses half of the base. Once the height is known, the ordinary triangle area formula applies.

Worked example

Equal sides = 10 Base = 12 Half base = 6 Height = √(10² - 6²) = √64 = 8 Area = 12 × 8 ÷ 2 = 48 Perimeter = 10 + 10 + 12 = 32

For angles, the calculator uses trigonometry after the height splits the triangle into right triangles. If your class has not started trigonometry yet, the side, height, area, and perimeter values are usually the main answers you need.

When the side lengths are impossible

The base must be shorter than the sum of the two equal sides. Since the equal sides are both a, the base must be less than 2a. If the base is exactly 2a, the triangle collapses into a straight line and has no area.

Common questions

  • If you know the equal side length and the base, first find the height by splitting the triangle into two right triangles. The height is √(a² - (b/2)²), where a is the equal side and b is the base. Then use area = base × height ÷ 2. This is why an isosceles triangle problem often becomes a right-triangle problem.
  • Yes. The angles opposite the equal sides are equal. If the two slanted sides are equal and the bottom side is the base, then the two angles along the base are equal. This property is often used to find missing angles without using trigonometry.
  • Yes, the base can be longer than each equal side individually, but it cannot be as long as or longer than the sum of the two equal sides. For equal sides of 10 and 10, the base must be less than 20. If it is 20 or more, the sides cannot close to form a real triangle.
  • Draw the altitude from the vertex angle to the base. In an isosceles triangle, that altitude cuts the base in half. Then use the Pythagorean theorem with the equal side as the hypotenuse and half the base as one leg. The remaining leg is the height.
  • Yes, depending on the definition used in your class. An equilateral triangle has three equal sides, so it certainly has at least two equal sides. Many textbooks treat equilateral triangles as a special type of isosceles triangle, although some classrooms separate the names for clarity.
  • In an isosceles triangle, the two equal sides create symmetry. The altitude from the vertex to the base is also a median and an angle bisector. That means it meets the base at its midpoint, making two congruent right triangles.