Do you want to solve mathematical and geometric problems of the Pythagorean relationship? Check out the online calculator to easily solve the Pythagorean theorem (Three Squares Theorem).
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This online application is used to calculate the Pythagorean theorem and find the value of the hypotenuse (c – longest side) in a right triangle. For the calculation, you just have to enter the lengths of the two legs (a and b – perpendicular sides) and click on calculate. The result will be displayed automatically.
The Pythagorean theorem is a fundamental formula in geometry that applies to right triangles, that is, triangles that have a 90° angle (right angle).
What is the pythagorean theorem?
The Pythagoras’ theorem states that in a right triangle, the sum of the squares of the legs (sides that form the right angle) is equal to the square of the hypotenuse (side opposite the right angle and longest). The Pythagoras equation is: a² + b² = c² (a and b are legs, and c is the hypotenuse)
- Legs: The two sides that form the right angle (perpendicular sides)
- Hypotenuse: The side opposite the right angle (longest side)
- Right triangle: The Pythagorean equation only applies to triangles that have a 90-degree angle.
How to calculate the Pythagorean theorem?
To apply Pythagoras rule and calculate the hypotenuse (c) of a right triangle. The forluma is: a² + b² = c². Then, from the Pythagorean equation, we can extract that c = √(a² + b²), where a and b are the legs that form the right angle.
Some calculations of Pythagorean theorem examples:
- Exercise 1. Imagine you have a right triangle where one leg measures 3 centimeters and the other measures 4 centimeters… What would be the value of the hypotenuse, the longest side? We could do the following calculations:
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- Square of the first leg: 3×3 = 9 cm (Value leg a)
- Square of the second leg: 4×4 = 16 cm (Value leg b)
- Now, we add those results: 9+16 = 25 cm (Value of hypotenuse)
- If we apply the Pythagorean law directly we would have the following: Hypotenuse = √ (a² + b²), that is, Hypotenuse = √ (3² + 9²) = 25 cm
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- Exercise 2: We have a right triangle, the hypotenuse measures 10 m and one leg measures 6 m… What is the value of the other leg (b)?
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- If we base it on the Pythagorean theorem (a² + b² = c²) and solve for b to perform the calculations, we find that leg b = √ (a² + c²)
- The calculations would be: b = √ (6² + 10²) = 8 cm the value of the leg.
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Remember that the Pythagorean theorem only applies to right triangles (also called the theorem of the three squares), but it is a fundamental tool in geometry, trigonometry, physics, engineering, construction and many scientific disciplines. Do you dare with the next exercise quiz?
Pythagorean Theorem Quiz
Exercise 1: Find the Hypotenuse
A right triangle has legs of 3 cm and 4 cm.
How long is the hypotenuse?
cm
Exercise 2: Find a Leg
A right triangle has a hypotenuse of 10 cm and one leg of 6 cm.
How long is the other leg?
cm
Exercise 3: Isosceles Right Triangle
A right triangle has two equal legs of 7 cm each.
How long is the hypotenuse? (round to 1 decimal place)
cm
Exercise 4: Practical Problem
A 5-meter ladder leans against a wall. If the base of the ladder is 3 meters from the wall, how high up the wall does it reach?
meters
Exercise 5: Diagonal of a Rectangle
A rectangle has dimensions of 12 cm × 9 cm.
What is the length of its diagonal?
cm
Exercise 6: Right Triangle Verification
Is a triangle with sides of 5, 12, and 13 units a right triangle?
Exercise 7: Distance Between Points
What is the distance between points A(1, 2) and B(4, 6) on a Cartesian plane?
(round to 1 decimal place)
units
Curiosity! The Gougu theorem (in East Asia): In some Asian cultures, especially Chinese, this theorem was discovered and formulated independently long before Pythagoras, and it is known by this name.
🟧 Note: Our Pythagorean theorem calculator also allows for inverse calculations if you know the hypotenuse and one of the legs, and you will also have a geometric representation of the obtained right triangle.
Pre-calculated examples of the Pythagorean theorem
Here is a table with some pairs of legs and their respective hypotenuses already calculated according to the Pythagoras theorem:
| Leg a (vertical) | Leg b (horizontal) | Hypotenuse c |
|---|---|---|
| 0.3 | 0.4 | 0.5 |
| 1 | 1 | ≈1.414 |
| 2 | 2 | ≈2.828 |
| 2.5 | 6 | ≈6.5 |
| 3 | 4 | 5 |
| 3.5 | 12 | ≈12.5 |
| 4.5 | 6 | ≈7.5 |
| 5 | 12 | 13 |
| 5.5 | 13.2 | ≈14.3 |
| 6 | 8 | 10 |
| 7 | 24 | 25 |
| 8 | 15 | 17 |
| 9 | 12 | 15 |
| 9 | 40 | 41 |
| 10 | 10 | ≈14.142 |
| 11 | 60 | 61 |
| 12 | 16 | 20 |
| 14 | 48 | 50 |
| 15 | 20 | 25 |
| 16 | 30 | 34 |
| 18 | 24 | 30 |
| 20 | 21 | 29 |
| 7.5 | 10 | ≈12.5 |
| 9.9 | 13.2 | ≈16.5 |
🟧 Any other questions about the calculator or calculations on the pythagoras and theorem (theorem of the three squares) you can consult us from the contact section in the footer.
Frequently asked questions about the Pythagorean theorem – FAQ
What is the Pythagorean theorem?
The Pythagorean theorem is a mathematical formula used to find the length of a side in a right triangle when the lengths of the other two sides are known. The equation is a² + b² = c², where “a” and “b” are the legs and “c” is the hypotenuse.
What is the Pythagorean theorem used for?
It is mainly used to solve problems with right triangles, such as finding the length of an unknown side. It also has practical applications in construction, navigation, design, and many areas of physics and engineering.
How is the Pythagorean theorem applied?
It is applied by squaring the lengths of the two legs, adding them, and then taking the square root of the result to obtain the hypotenuse. If a leg is sought, the square of the other known leg is subtracted from the square of the hypotenuse and then the square root is taken.
What is the formula of the Pythagorean theorem?
The formula is a² + b² = c², where a and b represent the legs of the right triangle and c represents the hypotenuse, which is the longest side and opposite the right angle.
How do you know if a triangle is right using the Pythagorean theorem?
If when applying the formula a² + b² = c² with the triangle’s measurements, the equality holds, then the triangle is right. If it does not hold, it is not.
Does the Pythagorean theorem work on any triangle?
No, it only works on right triangles, which have a 90-degree angle. If the triangle is not right, the formula does not apply.
How do you know which is the hypotenuse in a right triangle?
The hypotenuse is always the longest side of the right triangle and is opposite the 90-degree angle. The other two sides are the legs.
How to find the hypotenuse with the Pythagorean theorem?
To find the hypotenuse, the lengths of the two legs are squared, added, and then the square root is applied to the result. This way, we obtain the length of the hypotenuse.
How to find a leg with the Pythagorean theorem?
To find an unknown leg, the hypotenuse is squared, the square of the other known leg is subtracted, and then the square root of the result is applied.
Why is the Pythagorean theorem important?
It is important because it is a fundamental tool in geometry and trigonometry. It allows solving problems related to distances and angles and has multiple applications in science, technology, engineering, and art.
Who invented the Pythagorean theorem?
Although the theorem is named after the Greek philosopher Pythagoras, who lived around the 6th century BC, there is evidence that earlier civilizations, such as the Babylonians and Egyptians, already used this relationship between the sides of a right triangle.
What is a Pythagorean triangle?
It is a right triangle whose sides have lengths that are integers and satisfy a² + b² = c². A classic example is the triangle with sides 3, 4, and 5, because 3² + 4² = 9 + 16 = 25 = 5²
