1. Geometry Volume of Pyramids and Cones
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2. Find the unknown numbers.
Warm Up
Find the unknown numbers.
The difference of two numbers is 24. the large
number is 4 less than 3 times the smaller number.
2) Three times the first number plus the second number is 88. The first number times 10 is equal to 4 times the second.
3) The sum of two numbers is 197. The first number is 20 more than ½ of the second number.
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3. Volume of Pyramids and Cones
The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown.
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4. The square pyramids are congruent, so they have the same volume
The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.
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5. Volume of a Pyramid
The volume of a pyramid with base area B and height h is V = 1/3 Bh. h B
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6. Finding Volumes of Pyramids
Find the volume of each pyramid.
A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft.
b) The square pyramid
the base is a square with a side
length of 4 in., and the height is 6 in.
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7. Find the volume of each pyramid.
c) The trapezoidal pyramid with base ABCD, where
AB || CD and AE plane ABC.
9 m
6 m
10 m B E D C A
18 m
Step 1 Find the area of the base.
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8. 9 m
6 m
10 m B E D C A
18 m
Step 2 Use the base area and the height to find the volume. Because AE plane ABC, AE is the altitude, so the height is equal to AE.
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9. Find the volume of a regular hexagonal pyramid with a
Now you try!
Find the volume of a regular hexagonal pyramid with a
base edge length of 2 cm and a height equal to the area of the base.
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10. Architecture Application
The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft) 2
1 acre
10 stories
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11. 1 acre
10 stories
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12. 2) What would be the volume of the Rainforest Pyramid if
Now you try!
2) What would be the volume of the Rainforest Pyramid if
the height were doubled?
1 acre
10 stories
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13. Volume of a Cones h r
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14. Finding Volumes of a Cones
Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth.
A) A cone with radius 5 cm and height 12 cm
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15. B) A cone with a base circumference of 21 cm and a
height 3 cm less than twice the radius
Step 1: Use the circumference to find the radius.
Step 2: Use the radius to find the height.
2(10.5) – 3 = 18 cm The height is 3 cm less than twice
the radius.
Step 3: Use the radius and height to find the volume.
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16. Step 1: Use the Pythagorean Theorem to find the height.
25 ft
7 ft
Step 1: Use the Pythagorean Theorem to find the height.
Step 2: Use the radius and height to find the volume.
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17. 3) Find the volume of the cone.
Now you try!
3) Find the volume of the cone.
18 m
8 m
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18. Exploring Effects of Changing Dimensions
The length, width, and height of the rectangular
pyramid are multiplied by ¼ . Describe the effect on the volume.
20 ft
24 ft
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19. Length, width, and height multiplied by ¼: Original dimensions:
20 ft
24 ft
Length, width, and height multiplied by ¼:
Original dimensions:
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20. 4) The radius and height of the cone are doubled.
Now you try!
4) The radius and height of the cone are doubled.
Describe the effect on the volume.
9 cm
18 cm
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21. Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
2 in
4 in
5 in
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22. 5) Find the volume of the composite figure.
Now you try!
5) Find the volume of the composite figure.
15 ft
12 ft
25 ft
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23. Now some problems for you to practice !
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24. Assessment
1) Find the volume of each pyramid. Round to the nearest tenth, If necessary. A
17 in
6 in
4 in B
4 cm cm
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25. 2) A crystal is cut into the shape formed by two square pyramids joined at the base. Each pyramid has a base edge length of 5.7 mm and a height of 3 mm. what is the volume to the nearest cubic millimeter of the crystal?
3 mm
5.7 mm
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26. 3) Find the volume of each cone
3) Find the volume of each cone. Give your answer both in terms of and rounded to the nearest tenth.
14 cm
9 cm
30 in.
24 in. A B
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27. 4) Describe the effect of the each change on the
volume of the given figure.
3 cm
5 cm
a) The dimensions are tripled
b) The dimensions are
multiplied by ½.
15 ft
9 ft
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28. 5) Find the volume of each composite figure. Round to the
nearest tenth, if necessary.
12 cm
18 cm A B
4 in.
8 in.
6 in
12 in.
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29. Let’s review
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30. Volume of Pyramids and Cones
The volume of a pyramid is related to the volume of a prism with the same base and height. The relationship can be verified by dividing a cube into three congruent square pyramids, as shown.
Next Page:
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31. The square pyramids are congruent, so they have the same volume
The square pyramids are congruent, so they have the same volume. The volume of each pyramid is one third the volume of the cube.
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32. Volume of a Pyramid
The volume of a pyramid with base area B and height h is V = 1/3 Bh. h B
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33. Finding Volumes of Pyramids
Find the volume of each pyramid.
A rectangular pyramid with length 7 ft, width 9 ft, and height 12 ft.
b) The square pyramid
the base is a square with a side
length of 4 in., and the height is 6 in.
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34. Find the volume of each pyramid.
c) The trapezoidal pyramid with base ABCD, where
AB || CD and AE plane ABC.
9 m
6 m
10 m B E D C A
18 m
Step 1 Find the area of the base.
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35. 9 m
6 m
10 m B E D C A
18 m
Step 2 Use the base area and the height to find the volume. Because AE plane ABC, AE is the altitude, so the height is equal to AE.
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36. Architecture Application
The Rainforest Pyramid in Galveston, Texas, is a square pyramid with a base area of about 1 acre and a height of 10 stories. Estimate the volume in cubic yards and in cubic feet. (Hint: 1 acre = 4840 yd, 1 story = 10 ft) 2
1 acre
10 stories
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37. 1 acre
10 stories
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38. Volume of a Cones h r
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39. Finding Volumes of a Cones
Find the volume of each cone. Give your answers both in terms of and rounded to the nearest tenth.
A) A cone with radius 5 cm and height 12 cm
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40. B) A cone with a base circumference of 21 cm and a
height 3 cm less than twice the radius
Step 1: Use the circumference to find the radius.
Step 2: Use the radius to find the height.
2(10.5) – 3 = 18 cm The height is 3 cm less than twice
the radius.
Step 3: Use the radius and height to find the volume.
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41. Step 1: Use the Pythagorean Theorem to find the height.
25 ft
7 ft
Step 1: Use the Pythagorean Theorem to find the height.
Step 2: Use the radius and height to find the volume.
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42. Exploring Effects of Changing Dimensions
The length, width, and height of the rectangular
pyramid are multiplied by ¼ . Describe the effect on the volume.
20 ft
24 ft
Next Page:
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43. Length, width, and height multiplied by ¼: Original dimensions:
20 ft
24 ft
Length, width, and height multiplied by ¼:
Original dimensions:
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44. Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
2 in
4 in
5 in
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45. You did a great job today!
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