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Space Diagonal The Pythagorean Theorem in 3D

Space Diagonal

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Space Diagonal. The Pythagorean Theorem in 3D. In a rectangular prism, a space diagona l 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. - PowerPoint PPT Presentation

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Page 1: Space Diagonal

Space Diagonal

The Pythagorean Theorem in 3D

Page 2: 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.

A rectangular prism has 4 space diagonals.

aka, ‘stick in the box’

Page 3: 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.

Space Diagonal

Page 4: Space Diagonal

F

28cm

12cm

16cm

A B

CD

E

GH

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.

(1) Start by considering the triangle ABC.

(2) Calculate the length of AC.

AC ≈ 30.5cm

A

C

28

12

B22 1228 AC

Page 5: Space Diagonal

Now look at the box again.

F

28cm12cm

16cm

A B

CD

E

GH

30.5cm

(3) Now consider triangle ACG.

(4) Calculate the length of AG.

A

G

30.5

16

AG =

AG = 34.4cm

22 165.30

C

Page 6: Space Diagonal

OROR

OR

OR

OR

OR

OROR

OR

Page 7: Space Diagonal

A diagonal of a three-dimensional figure connectstwo vertices of two different faces.

Diagonal d of a rectangular prism is shown in the diagram.

Using substitution, d2 =l2 + w2 + h2.

By the Pythagorean Theorem, l2 + w2 = x2 and x2 + h2 = d2.

Page 8: Space Diagonal

Pythagorean Theorem2D vs 3D

c

b

d

22 cbd 222 cbah

Page 9: Space Diagonal

d2 = l2 + w2 + h2.

The Space Diagonal

Page 10: Space Diagonal

Space DiagonalPythagoras In 3D Shapes.

F

28cm12cm

16cm

A B

CD

E

GH

222 161228 AG

41.341184 AG

222 CGBCABAG

Page 11: Space Diagonal

What Goes In The Box ?

Calculate the lengths of the space diagonals in the rectangular prisms below:

(1)

4m3m

12m(2)

26cm 17cm

23cm

222 1234 d

md 13169

222 231726 d

cmd 65.381494

Page 12: Space Diagonal

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

Page 13: Space Diagonal

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.

225 = 144 + 49 + h2

h2 = 32

Substitute

Square both sides of the equation.

Simplify.

Solve for h2.

Take the square root of both sides.