7.2 3D Applications of the Pythag. Theorem
Now that we know how to use the Pythagorean Theorem to find either a missing leg or missing hypotenuse,
we can move this concept into three-dimensional concepts. Let’s look at some examples.
Regular Cones and Pyramids
In both cones and pyramids we can use the Pythagorean Theorem to find the height or slant height. We
can also find the radius in a cone or the side length of the base of a pyramid. First look at this cone.
slant
height
ℎ
7
H
The height and radius are fairly obvious, but the slant height might be new
vocabulary. The slant height, usually referred to as H in problems, is the height from the
outer bottom edge of the cone up to the tip. It is not the actual height because it is not
perpendicular to the base. Notice that each of these three variables form a right triangle.
Therefore if we know two of them we can find the other one.
For example, assume that 7 5 ;<. and ℎ = 12 ;<. What is H, or the slant height,
in inches?
? + @ = + 7 + ℎ = H 5 + 12 = H 25 + 144 = H 169 = H √169 = ”H 13 ;<. = H
Why is this useful? Knowing each dimension allows us to find the volume or surface area of the shape. For
example, the formula for the surface of a cone is ¼P = ½7 + ½7H. Now that we know H = 13 ;<. we can find the
surface area is ¼P = 25½ + 65½ = 90½ ;< .
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Let’s look at a pyramid missing its height.
slant
height
?
Assume that the base of the pyramid (bottom) is a square with side length of
12 +O. and the slant height is 10 +O. What is the height of the pyramid? Since we have
the side length of the square, we only need half of that to form the short leg of our right
triangle, or 6 +O. This gives us the following:
? + @ = + ? + ℎ = H 6 + ℎ = 10
36 + ℎ = 100
−36
− 36
ℎ = 64
”ℎ = √64
ℎ = 8 +O.
Again, we can now answer further questions about this shape like finding the volume. The volume of a
regular pyramid uses the formula ¾ = C Qℎ where Q is the area of the base shape (the square). This means that
the volume is C 1448 = 384 +OC .
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Lesson 7.2
Use the picture below to find information about the pyramid with a square base in problems 1-8. Round your
answers to three decimal places if necessary.
slant
height
?
H
1. The pyramid has a square base that is 70 ) on each side. The slant height is
37 ). What is ℎ, the height of the pyramid?
2. The pyramid has a square base that is 50 O on each side. The slant height is 30 O. What is ℎ, the height?
3. The pyramid has a square base that is 14 +O on each side. The height is 24 +O. What is H, the slant height?
4. The pyramid has a square base that is 70 OO on each side. The height is 10 OO. What is H, the slant height?
5. The height of the pyramid is 15 +O, and the slant height is 39 +O. Find the value of ? in the diagram.
6. The height of the pyramid is 80 ;<, and the slant height is 82 ;<. Find the value of ? in the diagram.
7. The slant height is 17 ) and the height is 8 ). What is J, the side length of the base?
8. The slant height is 50 ) and the height is 32 ). What is J, the side length of the base?
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Use the picture below to find information about the pyramid in problems 9-14. Round your answers to three
decimal places if necessary.
7
H
slant
height
9. The cone has a radius of 12 +O and a height of 5 +O. What is H, the slant height
of the cone?
10. The cone has a radius of 15 OO and a height of 8 OO. What is H, the slant height of the cone?
11. The cone has a radius of 30 ) and a slant height of 34 ). What is ℎ, the height of the cone?
12. The cone has a radius of 33 O and a slant height of 65 O. What is ℎ, the height of the cone?
13. The cone has a height of 16 +O and a slant height of 65 +O. What is 7, the radius of the cone?
14. The cone has a height of 4 ;< and a slant height of 6 ;<. What is 7, the radius of the cone?
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Use the picture below to find lengths of segments in the rectangular prism in problems 15-20. Round your
answers to three decimal places if necessary.
D
15. The length of ±±±±
PQ is 6 ) and the length of ±±±±
QR is 8 ). Find the length of ±±±±
PR .
C
A
B
16. The length of ±±±±
PQ is 23 +O and the length of ±±±±
QR is 70 +O. Find the length of ±±±±
PR .
,R is 84 OO. Find the length of ±±±±
P,.
17. The length of ±±±±
PR is 13 OO and the length of ±±±±
18. The length of ±±±±
PR is 11 OO and the length of ±±±±
,R is 30 OO. Find the length of ±±±±
P,.
19. The length of ±±±±
PQ is 4 ), the length of ±±±±
QR is 3 ) and the length of ±±±±
,R is 12 ). Find the length of ±±±±
P,.
20. The length of ±±±±
PQ is 2 ), the length of ±±±±
QR is 3 ) and the length of ±±±±
,R is 10 ). Find the length of ±±±±
P,.
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