space diagonal lesson
TRANSCRIPT
Space DiagonalThe Pythagorean Theorem in 3D
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’
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
F
28cm12cm
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
Now look at the box again.
F
28cm12cm
16cm
A B
CD
EGH
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
OROR
OR
OR
OR
OR
OROR
OR
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.
Pythagorean Theorem2D vs 3D
c
b
d
22 cbd 222 cbah
d2 = l2 + w2 + h2.The Space Diagonal
Space DiagonalPythagoras In 3D Shapes.
F
28cm12cm
16cm
A B
CD
E
GH
222 161228 AG41.341184 AG
222 CGBCABAG
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
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.
225 = 144 + 49 + h2
h2 = 32
Substitute
Square both sides of the equation.
Simplify.
Solve for h2.Take the square root of both sides.