1. a) Calculate the volume of the sphere.
V r 2572
3
The volume of the sphere is 2572 cm3. b) Calculate the volume of the sphere.
V r 248
3
The volume of the sphere is 248 475 cm3. 2. a) Calculate the volume of the outer sphere.
V r
Calculate the volume of the inner sphere.
V r 407
3
Subtract the volume of the inner sphere from the volume of the outer sphere.
V V V
838 778 720 431 058 cm
The volume of the space between the spheres is 1 431 058 cm3.
b) Convert the volume to capacity, 1 L equals 1000 cm3.
1431 058 ÷ 1000 ≈ 1431L
The capacity of the space between the two spheres is approximately 1431 L.
3. a) Use the formula for calculating the volume of a pyramid.
V lwh
The volume of the pyramid is 1615 in3. b) Use the formula for calculating the volume
of a pyramid.
V lwh
The volume of the pyramid is 40 ft3. c) Calculate the volume of the top pyramid.
V lwh
Calculate the volume of the bottom pyramid.
V lwh
Calculate the total volume.
V V V
The volume of the two pyramids is 154.5 cm3.
AlTErNATE SoluTioN
Students may have noticed that since the two pyramids share a base you could also have calculated the volume as if the two pyramids were just one big pyramid.
V lw h h
The volume of the two pyramids is 154.5 cm3. 4. Convert the units from feet to metres.
20 20 0 305
Calculate the volume of the pyramid.
V lwh
Calculate the volume of the rectangular prism.
V lwh
Calculate the total volume.
V V V
The volume of the figure is 137.6 m3. Convert from volume to capacity.
137 6. ×1000 = 137600 L
The capacity of the figure is 137 600 L.
AlTErNATE SoluTioN
If you know the conversion rates from imperial volume to capacity you could have also calculated the solution as follows using imperial measures.
Calculate the volume of the pyramid.
V lwh 880
=
Calculate the volume of the rectangular prism. 3960
=
= × ×
= ft
Calculate the total volume.
V V V
880 3960 4840 ft
The volume of the figure is 4840 ft3.
Convert from volume to capacity, 1 ft3 equals 7.48 US gallons.
4840 × 7 48. ≈ 36 203US gal
The capacity of the figure is 36 203 US gallons.
5. Calculate the volume of the pyramid.
V lwh
Calculate the volume of the rectangular prism.
V lwh
Calculate the difference in the two volumes.
x V V
5376 1792 3584in3
The difference in the volumes is 3584 in3.
6. a) Calculate the height of the pyramid using the Pythagorean theorem.
h
Calculate the volume of the pyramid.
V lwh
The volume of the pyramid is 7309 mm3. b) Calculate the height of the pyramid using
the Pythagorean theorem.
h
Calculate the volume of the pyramid.
V lwh
The volume of the pyramid is 2663 in3. c) Calculate the height of the pyramid using
the Pythagorean theorem.
h
Calculate the volume of the pyramid.
The volume of the pyramid is 637 in3.
d) Calculate the height of the pyramid using the Pythagorean theorem.
h 207 14 4
= −
( )
= −
=
≈ . mm
Calculate the volume of the pyramid.
V lwh
The volume of the pyramid is 806 mm3. 7. a) Calculate the volume of the cone.
V A h
The volume of the cone is 379.6 in3. b) Calculate the volume of the first cone.
V r h
Calculate the volume of the second cone.
V r h
Calculate the total volume.
V V V 1583
. .
cm
The total volume of the figure is approximately 1583 cm3.
c) Calculate the volume of the cone.
V r h
Calculate the volume of the cylinder.
V A h
V r h
V V
base 2
Calculate the total volume.
V V V 198
. .
m
The total area of the figure is 198 m3. 8. Use the formula for the volume of a cone to
solve for the height.
V r h
The height of the cone is 27 mm.
9. Use the Pythagorean theorem to calculate the height of the cone.
h 161 12 7
= −
= −
=
≈ . cm
Calculate the volume of the cone.
V r h 851
2
The volume of the cone is 851 cm3.
PrAcTicE Your SkillS, p. 192
1. a) Calculate the volume of the sphere.
V r
V V
sphere =
= ×
≈ 43
43 6 3 1047
3
The volume of the sphere is 1047 mm3. b) Calculate the volume of the cone.
V r h 515
2
The volume of the cone is 515 mm3. c) Calculate the volume of the pyramid.
V lwh
V V
pyramid =
= × × ×
The volume of the pyramid is 30.6 mm3.
d) Calculate the volume of the hemisphere.
V r
Calculate the volume of the cone.
V r h
Calculate the total volume.
V V V 271
. .
mm
The volume of the figure is 271 mm3. 2. Calculate the volume of the water tower.
V r
Convert the volume into capacity.
15 902 × 7.48 ≈ 118 947 US gal.
The capacity of the spherical water tower is 118 947 US gal.
3. a) Calculate the volume of the pyramid.
V lwh
Calculate the volume of the cone.
2309
2
Calculate the difference in volumes.
x x
= −
=
2940 2309 631cm3
The difference in volumes is 631 cm3. b) The difference in capacity is the difference
in volume converted to capacity. One centimetre cubed is equal to one millilitre.
The difference in capacity is 631 mL.
4. Calculate the volume of the pile of grain.
V r h
There are 6.4 m3 of grain in the pile.
5. Calculate the volume of the first sphere.
V r
Calculate the volume of the second sphere.
V r
Calculate the volume of the third sphere.
V r
Calculate the total volume.
V V V V 664
. . .
cm
The volume of water displaced is 664 cm3.
chAPTEr TEST, p. 194
1. a) Calculate the slant height of the triangle using the Pythagorean theorem.
h
Calculate the area of one of the triangles.
A bh
Calculate the area of the base.
A lw
Calculate the total surface area.
SA A A
The total surface area is 124.8 cm2.
b) Calculate the surface area of each face. 120
8 8
Calculate the total surface area.
SA A A 608
1 2
cm2
The total surface area is 608 cm2. c) Calculate the surface area of the
triangular face.
A bh
Calculate the slant height of the triangle using the Pythagorean theorem.
s
Calculate the area of each rectangular surface.
A 144
9 4 18
Calculate the total surface area.
SA A A A A 443
1 2 3 4
cm2
The surface area of triangular prism is 443 cm2.
2. Calculate the area of each face of the box.
A
Calculate the total surface area.
SA A A A
The surface area of the box is 87 m2. Calculate the volume of the box.
V lwh
The volume of the box is 45 m3. 3. a) Calculate the area of circular base.
SA r
Calculate the area of the lateral side of the can.
SA rh
The surface area of the can is 268.6 cm2. b) Calculate the volume of the can.
V A h
The volume of the can is 318.1 cm3. c) Use the volume of the can to calculate the
capacity of the can. Each centimetre cubed is equal to 1 mL so the capacity of the can is 318.1 mL.
4. a) Calculate the surface area of the ball.
SA r
The surface area of the ball is 39 408 mm2. b) Calculate the surface area of the ball.
V r 735 619
3
The volume of the sphere is 735 619 mm3. 5. Calculate the slant height the pyramid.
s
Calculate the area of the triangular faces of the pyramid.
A bh
Calculate the area of the lateral faces of the top prism.
A 176
= ×
= cm
Calculate the area of the large front face on the bottom.
A 350
= ×
= cm
Calculate the area of the side face on the bottom.
A 154
= ×
= cm
Calculate the area of the bottom of the figure.
A 275
= ×
= cm
Calculate the area of the space on top of the larger rectangle that is filled by the smaller figure.
A 121
= ×
= cm
Calculate the total surface area.
SA = 4A1 + 4A2 + 2A3 + 2A4 + 2A5 – 2A6 SA = 4 × 57.8 + 4 × 176 + 2 × 350 + 2 × 154 + 2 × 275 – 121
SA = 231.2 + 704 + 700 + 308 + 550 – 121 SA = 2372.2 cm2
The surface area of the figure is 2372.2 cm2.
Calculate the volume of the pyramid. 363
=
= × × ×
= cm
Calculate the volume of the top rectangular prism.
V lwh 1936
=
= × ×
= cm
Calculate the volume of the bottom rectangular prism.
V lwh 3850
=
= × ×
= cm
Calculate the total volume.
V V V V
363 1936 3850 6149 cm
The volume of the figure is 6149 cm3. 6. Calculate the volume of the pile.
V r h
The volume of the pile is 33.9 m3.
7. Calculate the area of one face of the pyramid.
A bh 108
=
= × ×
= m
Calculate the total surface area.
SA A 432
1
m2
Calculate the cost of the roofing material.
432 × 5 75. = $2484
The cost of the roofing material is $2484.00.