The Pythagorean Theorem in 3 D Space Diagonal

The Pythagorean Theorem in 3 D Space Diagonal

In a rectangular prism, a space diagonal is a line that goes from a vertex of the prism, through the center of the prism to the opposite vertex. aka, ‘stick in the box’ A rectangular prism has 4 space diagonals.

Space Diagonal The Pythagorean Theorem is useful when we need to find the length of a space diagonal in a rectangular prism. Side c is the space diagonal.

Finding The Length Of A Space Diagonal. Look at the rectangular prism below: It is our task to calculate the length of the space diagonal AG which runs through the space inside the rectangular prism. H G (1) Start by considering the triangle ABC. E F 16 cm (2) Calculate the length D of AC. C 12 cm A 28 cm C B A AC ≈ 30. 5 cm 12 28 B

Now look at the box again. H G E F D 28 cm (4) Calculate the length of AG. G 16 16 cm C 12 cm 30. 5 cm A (3) Now consider triangle ACG. A 30. 5 B AG = 34. 4 cm C

OR OR OR

A diagonal of a three-dimensional figure connects two vertices of two different faces. Diagonal d of a rectangular prism is shown in the diagram. By the Pythagorean Theorem, l 2 + w 2 = x 2 and x 2 + h 2 = d 2. Using substitution, d 2 =l 2 + w 2 + h 2.

Pythagorean Theorem 2 D vs 3 D c d b

The Space Diagonal 2 2 d = l + w + h.

Space Diagonal Pythagoras In 3 D Shapes. H G E F D A 16 cm C 12 cm 28 cm B

What Goes In The Box ? Calculate the lengths of the space diagonals in the rectangular prisms below: (1) (2) 23 cm 12 m 3 m 4 m 17 cm 26 cm

Using the Pythagorean Theorem in Three Dimensions Find the length of the space diagonal of a 6 cm by 8 cm by 10 cm rectangular prism Substitute Simplify

Using the Pythagorean Theorem in Three Dimensions Find the height of a rectangular prism with a 12 in. by 7 in. base and a space diagonal of 15 in. Substitute Square both sides of the equation. 225 = 144 + 49 + h 2 = 32 Simplify. Solve for h 2. Take the square root of both sides.