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CONFIDENTIAL 1 Geometry Geometry Volume of Pyramids Volume of Pyramids and Cones and Cones
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CONFIDENTIAL 1
GeometryGeometry
Volume of Pyramids Volume of Pyramids and Cones and Cones
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.
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:
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.
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
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
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:
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
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.
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.
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
CONFIDENTIAL 12
Now you try!
2) What would be the volume of the Rainforest Pyramid if the height were doubled?
1 acre
10 stories
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.
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.
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:
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.
CONFIDENTIAL 17
Now you try!
3) Find the volume of the cone.
18 m
8 m
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:
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.
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
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
CONFIDENTIAL 22
Now you try!
15 ft
12 ft
25 ft
5) Find the volume of the composite figure.
CONFIDENTIAL 23
Now some problems for you to practice !
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
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
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
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
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
CONFIDENTIAL 29
Let’s review
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:
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.
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
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
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:
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
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.
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
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.
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.
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:
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.
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:
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.
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
CONFIDENTIAL 45
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