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Learn the Pythagorean theorem easily with practical examples and photos. Prepare yourself for the exam.

One of the best known mathematical formulas is Pythagorean Theorem, which provides us with the relationship between the sides in a right triangle. A right triangle consists of two legs and a hypotenuse. The two legs meet at a 90° angle and the hypotenuse is the longest side of the right triangle and is the side opposite the right angle.

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1. Definition

A triangle has three sides. If you know the length of two of these sides, you can use it to calculate the length of the third side. Note: The triangle must have a right angle.

In the lower left corner there is a right angle. At these borders the sides a and b, which are called cathets. The longest side is c and is called the hypotenuse. The following formula is most commonly used in connection with the Pythagorean Theorem:

2. How to calculate it

In this section we first look at an example calculation for the Pythagorean theorem. In the second example there is a word problem to use the Pythagorean theorem.

Example 1: Calculate the length of the hypotenuse c


The cathets are 4 cm and 5 cm long. This means that a = 4 cm and b = 5 cm. We therefore change the formula to c and use both of these details. We calculate both squares and note that both the numbers and the units must be squared. We get by cm · cm = cm 2 . We summarize under the root and pull it. It should be noted that both the number and the unit must be the root. The root of cm 2 is therefore cm again.

Example 2: Pythagorean theorem word problem

In the second example we have a word problem (subject matter) for the Pythagorean theorem. The task: A ladder is leaned against a wall. The ladder is as long as the wall. The ladder is leaned in such a way that it lies 20 cm below the upper edge of the wall. The foot of the ladder is 1.20 m from the wall. How long is the ladder?


First we make a sketch. The wall is drawn in gray and the ladder in brown. The floor is still below. We know that the ladder and wall are the same height. We don't know how high, so we just write an x ​​on both of them. From the text of the task, we can see that the ladder on the ground is 1.20 meters from the wall. The distance between the top edge of the wall and the ladder is 20 cm, i.e. 0.2 m.

We can simplify the sketch to a triangle with a right angle. The right angle is at the bottom right. One cathete is 1.20 meters long. The hypotenuse is the longest side and opposite the right angle. We do not know the length, so we call it x. The cathete on the right is 20 centimeters shorter than the wall or ladder. Hence the length x minus 0.20 meters.

We now apply the Pythagorean Theorem. For this we take the general formula from above and adjust it.

We replace the c with x. The a is 1.20 m and the b becomes x - 0.2 m.


We can swap a and b, this makes no difference to the result.

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