# Using the Pythagorean Theorem in 3-Dimensional Shapes

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Using the Pythagorean Theorem in 3-Dimensional Shapes

Pythagorean Theorem Review The Pythagorean Theorem states: in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. a2 c b 2 c2 a b Find the length of AC in the diagram below: a2 52 132 a2 25 169 a2 144 a 12

3-Dimensional Figures 12 m 5m How would you find the length of segment AV? Do you see a right triangle inside the shape? How would you find the length of segment AF? Do you see the right triangle inside the shape?

3D Figures To Find the slant height of a cone or pyramid It doesn’t matter that it is 3D, there is a right triangle hidden in the problem The slant height is just the hypotenuse of a right triangle. Use a2 b2 c2 To Find the Diagonal of a 3D Prism Diagonals of prisms have a different formula (length)2 (width)2 (height)2 (diagonal of prism)2

Use the Pythagorean Theorem to find the length of diagonal AF. G AF is the diagonal going through a prism. To find AF, we use the Pythagorean Theorem 6c differently. m To find AF we need to know the length, width and height of the prism (because it is a 3D shape, we need 3 measurements) 6 cm What is the length AB? 2 What is the width FG? 3 cm What is the height GB? (length)2 (width)2 cm (height)2 (diagonal of prism)2 62 22 32 d2 36 4 9 d2 49 d2 The length of the diagonal AF is 7 d 7 cm.

Use the Pythagorean Theorem to find the length of diagonal TX. TX is the diagonal going through a prism. To find TX, we use the Pythagorean Theorem differently. What is the length TU? 12 in What is the width XW? 6 in What is the height UW? 9 in (length)2 (width)2 (height)2 (diagonal of prism)2 122 62 92 d2 144 36 81 d2 The length of the diagonal TX is 261 d2 which is between 16 and 17 d inches.