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Sample Problem Using Method 2. Use the length,width, and height of the rectangular solid with the Pythagorean Theorem to find the diagonal length.
Keywords: Pythagorean Theorem, Prism, Diagonal Length
Common Core Standard   8.G.B.7  8th Grade Math

### Find the length of the diagonal (AB) of a rectangular box.

5
C
7
B
A

Step 1. Use, Length^2 +Width^2 +Height^2 =Diagonal Length^2

5^2 +5^2 + 7^2 =AB^2
25 +25 + 49 = AB^2
99 =√AB
√99 = √AB
9.94 = Length of AB

Find the length of a diagonal of a rectangular prism with a side length of 4 units,width of 4 units and height 6 units.

A cube is a rectangular prism with six equal sides. As a result you can find the diagonal length the same way as a rectangular prism.

Method 1
Step 1. Find the diagonal length of the base.
4^2 +4^2 = c^2
16 +16 = c^2
​√32 =√c
5.65 = c

Step 2. Plug the diagonal length, and the height into the Pythagorean Theorem for the diagonal length of the cube.

5.65^2 + 6^2 = c^2
32 + 36 = c^2
√68 = √c
8.246 = Inner diagonal length

Method 2
Use, Length^2 +Width^2 +Height^2 =Diagonal Length

4^2 +4^2 +6^2 = Inner Diagonal Length^2
16+16+36 = Inner Diagonal Length^2
68 = Inner Diagonal Length^2
Take the square root both sides.
√ =68 √Inner Diagonal Length
8.246 = Inner Diagonal Length

Use this diagonal length calculator to check your work.

There are two methods you can use to find the length of the inner diagonal of a rectangular prism.

• Method one uses the length of the base diagonal, and the height of the rectangular solid to find the inner diagonal length.

• Method two uses the height,width, and length of the prism to find the diagonal length.

​Example Problem Using Method 1. B
A
Diagonal of a Rectangular Prism
A prism is a three dimensional figure with six sides, and two parallel rectangular bases.

What does diagonal length mean?
diagonal is a line segment that links two vertices, but the vertices are non-adjacent. (not next to each other). The diagonal length will be the measure of the distance from one vertices to another non-adjacent vertices of the rectangular prism.
See an example of an inner diagonal in the following picture. It is labeled  AB

### Diagonal Length Rectangular Prism            # Diagonal Length Rectangular Prism

Hi welcome to MooMooMath. Today we are going to find the diagonal length of a cube. What distance are we actually looking at? Well I’m going to draw a diagonal across the face of the cube. So I will go from A to B How do I find the distance across the top of the cube? First we will isolate this triangle right here and use the Pythagorean Theorem right here. I have a triangle here and I’m just pulling it out and it is 6 by 6 on each leg and we have a right triangle so we have a right triangle. So this triangle is a 45- 45-90 so you can use the Pythagorean Theorem A squared plus B squared equals C squared or you can use your short cut rules for 45-45-90 for a 45-45-90 if you know the legs to get the hypotenuse all you do is multiply it by the square root of two. So since these two legs are 6 I know my diagonal is 6 square root 2. I will go back and label the diagonal 6 square root two and you have the diagonal. Now let's find the distance of diagonal from A to C. You can find that distance from corner to corner inside a cube by again using the Pythagorean Theorem. I will highlight three sides that we are going to use. We will use the diagonal on the top, we will use the original height and then solve for the diagonal distance from corner to corner on that triangle. So I will use 6 square root two for my A squared and the shorter leg is 6 which is the height of the cube. And then I’m going to solve for my “c” which is my “c” squared. So see how I’m just plugging it into the Pythagorean Theorem- “a squared” plus “b squared” equals “c squared. Where “a” is this diagonal, “b” is the height, and “c” is the diagonal distance through. Ok now let’s multiply all this out. 36 times 2 is 72, plus 36 and that adds up to 108. Now I’ve got to find the square root of 108 and that’s just going to simplify to what, 54, times 2, which is 9 times 8 so 3 square root 16.

Step 1. Find the diagonal length of the base using the Pythagorean Theorem. Let's call this (A)

Step 2. Use this length of the base ( A ) with the height, and the Pythagorean Theorem to find the diagonal length of the 3D figure.  Video explains the basics of sine,cosine, and tangent? 