Space Diagonal

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Space Diagonal

• The Pythagorean Theorem in 3D

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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.
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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.
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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
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Now look at the box again.
(3) Now consider triangle ACG.
(4) Calculate the length of AG.
C
AG AG 34.4cm
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OR
OR
OR
OR
OR
OR
OR
OR
OR
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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, l2
w2 x2 and x2 h2 d2. Using substitution,
l2 w2 h2 d2.
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Pythagorean Theorem2D vs 3D
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The Space Diagonal
l2 w2 h2 d2.
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Space DiagonalPythagoras In 3D Shapes.
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What Goes In The Box ?
Calculate the lengths of the space diagonals in
the rectangular prisms below
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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
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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.
Simplify.
225 144 49 h2
Solve for h2.
h2 32
Take the square root of both sides.