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Isosceles Triangle Calculator

Calculate the area, perimeter, and height of an isosceles triangle from its equal side and base.

Reviewed by Chase FloiedUpdated

This free online isosceles triangle calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Length of the two equal sides

Length of the base (unequal side)

Results

Area

36.6606 sq units

Perimeter

28 units

Height

9.1652 units

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Isosceles Triangle Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Isosceles Triangle Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Isosceles Triangle Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

  • •Use the Isosceles Triangle Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Isosceles Triangle Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Calculate the area, perimeter, and height of an isosceles triangle from its equal side and base. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Isosceles Triangle Calculator

An isosceles triangle is a triangle with two sides of equal length. The angles opposite the equal sides are also equal, a property known as the isosceles triangle theorem (proved by Euclid in his Elements). Isosceles triangles appear frequently in architecture, design, and nature. The gables of houses, the cross-section of A-frame buildings, and many structural trusses are isosceles triangles. This calculator computes the height, area, and perimeter when you provide the length of the equal sides and the base. The height is found by dropping a perpendicular from the apex to the base, which bisects the base, creating two right triangles. This allows us to use the Pythagorean theorem: h = √(a² − (b/2)²).

The Math Behind It

An isosceles triangle has a line of symmetry along the altitude from the apex to the midpoint of the base. This altitude bisects the base and the apex angle, creating two congruent right triangles. Each right triangle has hypotenuse a (the equal side), one leg b/2 (half the base), and the other leg h (the height). By the Pythagorean theorem, h = √(a² − (b/2)²). The base angles are equal: β = arccos(b/(2a)). The apex angle is α = 180° − 2β. The area can be expressed as A = (b/4)√(4a² − b²). An equilateral triangle is a special isosceles triangle where the base also equals the equal sides. For an isosceles right triangle (45-45-90), the two legs are equal and the hypotenuse is a√2. The incircle radius is r = A/s where s is the semi-perimeter, and the circumcircle radius is R = a²/(2h) = a²/√(4a² − b²).

Formula Reference

Height of Isosceles Triangle

h = √(a² − (b/2)²)

Variables: a = equal side, b = base

Area

A = ½ × b × h

Variables: b = base, h = height

Worked Examples

Example 1: Roof cross-section

A roof has equal rafters of 5 m and a span (base) of 8 m.

Step 1:h = √(5² − (8/2)²) = √(25 − 16) = √9
Step 2:h = 3 m
Step 3:A = ½ × 8 × 3 = 12 m²

Height = 3 m, Area = 12 m²

Common Mistakes & Tips

  • !Confusing the equal side with the base when the triangle is oriented differently.
  • !Forgetting to halve the base when computing the height.
  • !Trying to compute the height when b > 2a (which is impossible — the sides cannot form a triangle).

Related Concepts

Equilateral Triangle

A special isosceles triangle where all three sides are equal.

Pythagorean Theorem

Used to find the height of the isosceles triangle.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Equilateral Triangle Calculator

Frequently Asked Questions

Can an isosceles triangle be a right triangle?

Yes. An isosceles right triangle has two 45° angles and one 90° angle. The two legs are equal, and the hypotenuse is leg × √2.

What if the base is longer than the equal sides?

That is fine as long as the triangle inequality holds: b < 2a. The triangle will be obtuse.