Geometry Volume of Pyramids and Cones
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CONFIDENTIAL 1
GeometryGeometry
Volume of Pyramids Volume of Pyramids and Cones and Cones
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CONFIDENTIAL 2
Warm UpWarm Up
Find the unknown numbers.
1) 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.
1)14 and 382) 16 and 403) 79 and 118
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CONFIDENTIAL 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.
Next Page:
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CONFIDENTIAL 4
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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CONFIDENTIAL 5
Volume of a Pyramid
The volume of a pyramid with base area B and height h is V = 1/3 Bh.
h h
BB
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CONFIDENTIAL 6
Finding Volumes of Pyramids
Next Page:
4 in.
4 in.
6 in.
Find the volume of each pyramid.
a)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.
V = 1
3Bh =
1
3(42)(6) = 32 in3
V = 1
3Bh =
1
3(7 9)(12) = 252 ft3
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CONFIDENTIAL 7
Find the volume of each pyramid.
9 m
6 m
10 m B
E
D C
A
18 m
c) The trapezoidal pyramid with base ABCD, where AB || CD and AE plane ABC.
Step 1 Find the area of the base.
B = 1
2(b1 + b2)h Area of a trapezoid
=1
2(9 + 18)6 Substitute 9 for b1m 18 for b2, and 6 for h.
= 81 m2 Simplify.
Next Page:
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CONFIDENTIAL 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.
V = 1
3Bh Volume of a pyramid
= 1
3(81)(10) Substitute 81 for B and 10 for h.
= 270 m3
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CONFIDENTIAL 9
Now you try!
1) 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.
1) 108 cm3
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CONFIDENTIAL 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 storiesThe base is a squarewith an area of about4840 yd2. the base edgelength is 4840 = 70 yd.the height is about10(10) = 100 ft, orabout 33 yd.
Next Page:
First find the volume in cubic yards.
V = 1
3Bh Volume of a regular pyramid
= 1
3(702)(33) = 53,900 yd3 Substitute 702 for B and 33 for h.
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CONFIDENTIAL 11
1 acre
10 stories
Then convert your answer to find the volume in cubic feet. thevolume of one cubic yard is (3 ft)(3 ft)(3 ft) = 27 ft3.
Use the conversion factor 27 yd3
1 yd3 to find the volume in cubic feet.
53,900 yd3 27 yd3
1 yd3 1,455,300 ft3
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CONFIDENTIAL 12
Now you try!
2) What would be the volume of the Rainforest Pyramid if the height were doubled?
2) The volume will be doubled.
1 acre
10 stories
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CONFIDENTIAL 13
Volume of a Cones
h
r
h
r
The volume of a cone with base area B, radius r,
and height h is V = 1
3Bh, or V =
1
3r2h.
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CONFIDENTIAL 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
Next Page:
V =1
3r2h Volume of a cone
=1
3(5)2 (12) Substitute 5 for r and 12 for h.
= 100 cm3 314.2 cm3 Simplify.
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CONFIDENTIAL 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.
2r = 21 Substitute 21 for C. r = 10.5 cm Divide both sides by 2.
V = 1
3 r2h Volume of a cone
= 1
3(10.5)2 (18) Substitute 10.5 for r and 18 for h.
=661.5 cm3 2078.2 cm3 Simplify.Next Page:
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CONFIDENTIAL 16
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.
V = 1
3 r2h Volume of a cone
= 1
3(7)2 (24) Substitute 7 for r and 24 for h.
=392 ft3 1231.5 ft3 Simplify.
72 + h2 = 252 Pythgorean Theorem h2 = 576 Subtract 72 from both sides. h = 24 Take the square root of both sides.
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CONFIDENTIAL 17
Now you try!
3) Find the volume of the cone.
18 m
8 m
3) 648∏ m3
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CONFIDENTIAL 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 ft20 ft
Next Page:
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CONFIDENTIAL 19
20 ft
24 ft20 ft
Length, width, and height multiplied by ¼:
Original dimensions:
V = 1
3Bh
= 1
3(24 20)(20)
= 3200 ft3
V = 1
3Bh
= 1
3(6 5)(5)
= 50 ft3
Notice that 50 = 1
64(3200). I f the length, width, and height
are multiplied by 1
4, the volume is multiplied by
1
4 3
, or 1
64.
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CONFIDENTIAL 20
Now you try!
4) The radius and height of the cone are doubled. Describe the effect on the volume.
9 cm
18 cm
4) The volume will increase by 8 times.
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CONFIDENTIAL 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
The volume of the cylinder is V = r2h = 2 2 (2) = 8 in3.The volume of the cone is
V= 1
3r2h =
1
3 2 2(3) = 4 in3.
The volume of the composite figure isthe sum of the volumes. V= 8 + 4 = 12 in3 37.7 in3
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CONFIDENTIAL 22
Now you try!
15 ft
12 ft
25 ft
5) Find the volume of the composite figure.
5) 3000 ft3.
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CONFIDENTIAL 23
Now some problems for you to practice !
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CONFIDENTIAL 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
4 3 cm
1a) 136 in3.1b) 96 cm3
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CONFIDENTIAL 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
2) 64.98 mm3.
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CONFIDENTIAL 26
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
3a) 378∏ cm3
3b) 1440∏ in3.
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CONFIDENTIAL 27
4) Describe the effect of the each change on the volume of the given figure.
b) The dimensions are multiplied by ½.
15 ft
9 ft
9 ft
3 cm
5 cm
a) The dimensions are tripled
4a) volume increases by 27 times.4b) volume decreases by 8 times.
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CONFIDENTIAL 28
5) Find the volume of each composite figure. Round to the nearest tenth, if necessary.
B
4 in. 8 in.
6 in
12 in.12 cm
12 cm
12 cm
18 cm
A
5a) 2592 cm3.5b) 1920∏ in3.
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CONFIDENTIAL 29
Let’s review
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CONFIDENTIAL 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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CONFIDENTIAL 31
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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CONFIDENTIAL 32
Volume of a Pyramid
The volume of a pyramid with base area B and height h is V = 1/3 Bh.
h h
BB
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CONFIDENTIAL 33
Finding Volumes of Pyramids
Next Page:
4 in.
4 in.
6 in.
Find the volume of each pyramid.
a)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.
V = 1
3Bh =
1
3(42)(6) = 32 in3
V = 1
3Bh =
1
3(7 9)(12) = 252 ft3
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CONFIDENTIAL 34
Find the volume of each pyramid.
9 m
6 m
10 m B
E
D C
A
18 m
c) The trapezoidal pyramid with base ABCD, where AB || CD and AE plane ABC.
Step 1 Find the area of the base.
B = 1
2(b1 + b2)h Area of a trapezoid
=1
2(9 + 18)6 Substitute 9 for b1m 18 for b2, and 6 for h.
= 81 m2 Simplify.
Next Page:
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CONFIDENTIAL 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.
V = 1
3Bh Volume of a pyramid
= 1
3(81)(10) Substitute 81 for B and 10 for h.
= 270 m3
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CONFIDENTIAL 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 storiesThe base is a squarewith an area of about4840 yd2. the base edgelength is 4840 = 70 yd.the height is about10(10) = 100 ft, orabout 33 yd.
Next Page:
First find the volume in cubic yards.
V = 1
3Bh Volume of a regular pyramid
= 1
3(702)(33) = 53,900 yd3 Substitute 702 for B and 33 for h.
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CONFIDENTIAL 37
1 acre
10 stories
Then convert your answer to find the volume in cubic feet. thevolume of one cubic yard is (3 ft)(3 ft)(3 ft) = 27 ft3.
Use the conversion factor 27 yd3
1 yd3 to find the volume in cubic feet.
53,900 yd3 27 yd3
1 yd3 1,455,300 ft3
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CONFIDENTIAL 38
Volume of a Cones
h
r
h
r
The volume of a cone with base area B, radius r,
and height h is V = 1
3Bh, or V =
1
3r2h.
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CONFIDENTIAL 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
Next Page:
V =1
3r2h Volume of a cone
=1
3(5)2 (12) Substitute 5 for r and 12 for h.
= 100 cm3 314.2 cm3 Simplify.
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CONFIDENTIAL 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.
2r = 21 Substitute 21 for C. r = 10.5 cm Divide both sides by 2.
V = 1
3 r2h Volume of a cone
= 1
3(10.5)2 (18) Substitute 10.5 for r and 18 for h.
=661.5 cm3 2078.2 cm3 Simplify.Next Page:
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CONFIDENTIAL 41
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.
V = 1
3 r2h Volume of a cone
= 1
3(7)2 (24) Substitute 7 for r and 24 for h.
=392 ft3 1231.5 ft3 Simplify.
72 + h2 = 252 Pythgorean Theorem h2 = 576 Subtract 72 from both sides. h = 24 Take the square root of both sides.
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CONFIDENTIAL 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 ft20 ft
Next Page:
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CONFIDENTIAL 43
20 ft
24 ft20 ft
Length, width, and height multiplied by ¼:
Original dimensions:
V = 1
3Bh
= 1
3(24 20)(20)
= 3200 ft3
V = 1
3Bh
= 1
3(6 5)(5)
= 50 ft3
Notice that 50 = 1
64(3200). I f the length, width, and height
are multiplied by 1
4, the volume is multiplied by
1
4 3
, or 1
64.
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CONFIDENTIAL 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
The volume of the cylinder is V = r2h = 2 2 (2) = 8 in3.The volume of the cone is
V= 1
3r2h =
1
3 2 2(3) = 4 in3.
The volume of the composite figure isthe sum of the volumes. V= 8 + 4 = 12 in3 37.7 in3
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CONFIDENTIAL 45
You did a You did a greatgreat job job today!today!