Last updated: Jun 10, 2024

Isosceles Triangle Side Calculator

Q: How do I calculate the third side of an isosceles triangle?

If the only unknown side is the base side b , our steps depend on what information we have. Usually, we have the leg length a , the base angle α , and the vertex angle β . We can then apply trigonometry to one half of the isosceles triangle, which is a right triangle with angles α and β/2 , a hypotenuse of length a , and a side of length b/2 . From there, we can calculate b/2 and then b .

Q: How do I calculate the sides of an isosceles right triangle from its hypotenuse?

An isosceles right triangle has a vertex angle of β = 90° and base angle α = 45° . We can use trigonometry to calculate the remaining side because it's a right triangle, and we know its angles. We want to calculate the side length a , and we have the hypotenuse b . So, we can easily say that a = b × cos(45°) .

Q: What is the length of the equal sides of an isosceles right triangle with hypotenuse 20 cm?

Because we know an isosceles right triangle's angles to be 45° , 45° , and 90° , we can work out the equal side lengths a with trigonometry. Calculate the cosine of 45° as: cos(45°) = 1 / √2 = 0.7071 As the cosine represents the ratio of the adjacent side a to the hypotenuse b , we can say that: cos(45°) = a / b Make a the subject and complete the equation: a = b × cos(45°) = 14.14 cm

Created by Rijk de Wet

Rijk de Wet

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Holds a Master's degree in Data Science / Industrial Engineering, obtained at Stellenbosch University, South Africa. His research investigates using swarm intelligence to solve data clustering and other optimization problems. Will never refuse an offer to play some board games. See full profile

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Reviewed by Steven Wooding

Steven Wooding

Physicist holding a 1st class degree and a member of the Institute of Physics. Creator of the UK vaccine queue calculator, and featured in many publications, including The Sun, Daily Mail, Express, and Independent. Tenacious in researching answers to questions and has an affection for coding. Hobbies include cycling, walking, and birdwatching. You can find him on Twitter @OmniSteve. See full profile

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Table of contents

How do I use the isosceles triangle side calculator?What is an isosceles triangle?How do I calculate the third side of an isosceles triangle?Related calculators FAQs

Welcome to the isosceles triangle side calculator, where we'll learn all there is to know about an isosceles triangle's sides. In this article, we explain:

What an isosceles triangle is;
How to calculate the third side of an isosceles triangle; and
How to calculate the sides of an isosceles right triangle from the hypotenuse.

How do I use the isosceles triangle side calculator?

Using the isosceles triangle side calculator is as easy as counting to three! All you need to do is:

Enter the known dimensions of your isosceles triangle. These can be its angles, height, or even a side if you know it.
Find your results filled into their respective fields.
The calculator can work backward, too: fill in the sides, and the calculator will work out all the other dimensions.

If you'd like to learn how the side lengths are calculated, read on!

What is an isosceles triangle?

An isosceles triangle is a triangle with two equally long sides (which we call the legs) and are both denoted with a. The remaining side is denoted by b and is unique. The angles adjacent to b (which we call the base angles α) are also equal due to the legs being equal in length. The remaining angle at the top is called the vertex angle β and is unique.

An isosceles triangle with its sides, angles and heights marked. — An isosceles triangle. Note that the legs marked `a` are the same length, and the corners marked `α` are the same angle.

How do I calculate the third side of an isosceles triangle?

If the only unknown side is the base side b, our steps depend on what information we have. Usually, we have the leg length a, the base angle α, and the vertex angle β. We can then apply trigonometry to one half of the isosceles triangle, which is a right triangle with angles α and β/2, a hypotenuse of length a, and a side of length b/2. From there, we can calculate b/2 and then b.

FAQs

How do I calculate the sides of an isosceles right triangle from its hypotenuse?

An isosceles right triangle has a vertex angle of β = 90° and base angle α = 45°. We can use trigonometry to calculate the remaining side because it's a right triangle, and we know its angles. We want to calculate the side length a, and we have the hypotenuse b. So, we can easily say that a = b × cos(45°).

What is the length of the equal sides of an isosceles right triangle with hypotenuse 20 cm?

Because we know an isosceles right triangle's angles to be 45°, 45°, and 90°, we can work out the equal side lengths a with trigonometry.

Calculate the cosine of 45° as:
cos(45°) = 1 / √2 = 0.7071
As the cosine represents the ratio of the adjacent side a to the hypotenuse b, we can say that:
cos(45°) = a / b
Make a the subject and complete the equation:
a = b × cos(45°) = 14.14 cm