Sign up for your FREE 7-day trial.Get instant access to all the benefits of SparkNotes PLUS! Cancel within the first 7 days and you won't be charged. We'll even send you a reminder.

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Step 2 of 4

Choose Your Plan

Step 3 of 4

Add Your Payment Details

Step 4 of 4

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Welcome to PLUS!Let us show you around so you can easily find all of your new PLUS study tools.

Welcome to PLUS!

Let us show you around so you can easily find all of your new PLUS study tools.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

One of the most interesting and well-known formulas in math is the Pythagorean
Theorem, which only holds true for right triangles. The formula says that
the length of the hypotenuse squared is equal to the sum of the squares of
the lengths of the legs:

c^{2} = a^{2} + b^{2}

If a triangle is right, then this formula holds true. Conversely, if this
formula holds true, then you know that the triangle for which it works is a
right triangle. With this formula, given any two sides of a right triangle, you
can calculate the length of the third side. One way to picture it can be seen below.
In the figure above, the area of the square
with side length c is equal to the sum of the areas of the squares with side
lengths a and b. This is a physical interpretation of the Pythagorean Theorem.

The three sides of a right triangle can be of any length, provided that they
obey the laws of the Pythagorean Theorem. However, only certain groups of three
integers can be the lengths of a right triangle. Such groups of integers
are called Pythagorean triples. Some common
Pythagorean triples include , , , , and . Any multiple of
one of these groups of numbers also can be a Pythagorean triple. For example,
=3. You can check any of them for yourself. All of these
groups of three integers, these Pythagorean triples, satisfy the Pythagorean
Theorem.

The key application of the Pythagorean Theorem is the technique for calculating
the length of the third side given two of the sides' lengths and the fact that
the triangle is right. Knowing the lengths of sides of a triangle is handy in
deciding whether triangles are related.