Using the Pythagorean Theorem in 3-Dimensional Shapes

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Presentation transcript:

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 + b2 = c2 Find the length of AC in the diagram below: c a b a2 + 52 = 132 a2 + 25 = 169 a2 = 144 a = 12

3-Dimensional Figures 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?

To Find the Diagonal of a 3D Prism 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. AF is the diagonal going through a prism. To find AF, we use the Pythagorean Theorem differently. 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) What is the length = AB? What is the width = FG? What is the height = GB? (length)2 + (width)2 + (height)2 = (diagonal of prism)2 62 + 22 + 32 = d2 36 + 4 + 9 = d2 49 = d2 7 = d 6cm 6 cm 2 cm 3 cm The length of the diagonal AF is 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? What is the width = XW? What is the height = UW? (length)2 + (width)2 + (height)2 = (diagonal of prism)2 122 + 62 + 92 = d2 144 + 36 + 81 = d2 261 = d2 261 = d 12 in 6 in 9 in The length of the diagonal TX is 𝟐𝟔𝟏 which is between 16 and 17 inches.