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# Pythagorean Theorem in 3D

#### Grade 8 Math Worksheets

The Pythagorean Theorem states that the square of the length of the longest side (the hypotenuse) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

• Pythagorean Theorem in 3D
• Solved Examples
• FAQs

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## Pythagorean Theorem in 3D

The Pythagorean Theorem can be extended to three-dimensional space, where it is called the Pythagorean Theorem in 3D. In 3D space, the theorem states that the square of the length of the longest side (the hypotenuse) of a right triangle is equal to the sum of the squares of the lengths of the other two sides.

In mathematical notation, the Pythagorean Theorem in 3D can be written as:

c^2 = a^2 + b^2 + h^2

where c is the length of the hypotenuse, and a, b, and h are the lengths of the other three sides of the right triangle.

In 3D space, the sides of a right triangle are called legs, and the longest side (opposite the right angle between the height and diagonal of the base) is called the hypotenuse, just as in 2D space. The Pythagorean Theorem in 3D can be used to find the length of the hypotenuse or the length of one of the legs of a right triangle in 3D space.

The Pythagorean Theorem in 3D has many applications in fields such as physics, engineering, and computer graphics, where it is used to calculate distances, lengths, and angles in 3D space.

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## Solved Examples

Example 1:

Find the distance between two points in 3D space with coordinates (1, 2, 3) and (4, 5, 6).

Solution:

We can model this situation as a right triangle in 3D space, with the distance between the two points as the hypotenuse, and the differences in x, y, and z coordinates as the legs. Using the Pythagorean Theorem in 3D, we have:

c^2 = a^2 + b^2 + h^2

c^2 = (4 – 1)^2 + (5 – 2)^2 + (6 – 3)^2

c^2 = 3^2 + 3^2 + 3^2

c^2 = 27

c = sqrt(27)

Therefore, the distance between the two points is approximately 5.196 units.

Example 2:

A rectangular box has dimensions of 3 feet by 4 feet by 5 feet. What is the length of the longest diagonal of the box?

Solution:

We can model this situation as a right triangle in 3D space, with the diagonal of the box as the hypotenuse, and the dimensions of the box as the legs. Using the Pythagorean Theorem in 3D, we have:

c^2 = a^2 + b^2 + h^2

c^2 = 3^2 + 4^2 + 5^2

c^2 = 9 + 16 + 25

c^2 = 50

c = sqrt(50)

Therefore, the length of the longest diagonal of the box is approximately 7.071 feet.

## FAQs

##### Can the Pythagorean Theorem in 3D be used for non-right triangles?

No, the Pythagorean Theorem in 3D only applies to right triangles in 3D space. For non-right triangles in 3D space, other methods such as the Law of Cosines or Law of Sines are used to find the lengths of the sides.

##### Can the Pythagorean Theorem in 3D be used for any three sides of a triangle?

No, the Pythagorean Theorem in 3D can only be used for right triangles in 3D space, where one of the angles is a right angle. In a non-right triangle in 3D space, there is no hypotenuse, and the Pythagorean Theorem in 3D does not apply.

##### What are some practical applications of the Pythagorean Theorem in 3D?

The Pythagorean Theorem in 3D has many practical applications in fields such as architecture, engineering, and physics. For example, it can be used to calculate the distance between two points in 3D space, the length of the diagonal of a rectangular box, or the distance between a point and a plane.

##### How do you use the Pythagorean Theorem in 3D?

To use the Pythagorean Theorem in 3D, you need to identify the right triangle in 3D space and its three sides. Then, you can plug in the lengths of the two legs and the hypotenuse into the equation c^2 = a^2 + b^2 + h^2 and solve for the unknown length.

##### How is the Pythagorean Theorem in 3D related to the Pythagorean Theorem in 2D?

The Pythagorean Theorem in 3D is an extension of the Pythagorean Theorem in 2D, which applies to right triangles in a two-dimensional plane. Both theorems relate the lengths of the sides of a right triangle, where the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Gloria Mathew writes on math topics for K-12. A trained writer and communicator, she makes math accessible and understandable to students at all levels. Her ability to explain complex math concepts with easy to understand examples helps students master math. LinkedIn

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