Math worksheet titled "Volume Word Problems" featuring six geometry questions related to calculating volumes and surface areas of various 3D shapes like pyramids, cones, spheres, and cylinders.
Volume Word Problems worksheet with six geometry questions involving pyramids, cones, spheres, and cylinders, including calculations for volume and surface area.
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Step-by-step solution for: Volume Word Problems Worksheet | PDF Printable Geometry Worksheet
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Show Answer Key & Explanations
Step-by-step solution for: Volume Word Problems Worksheet | PDF Printable Geometry Worksheet
Here are the step-by-step solutions for each problem on the worksheet.
1) Calculate the volume of a pyramid with a square base of sides 7 cm and a vertical height of 11 cm.
* Step 1: Find the area of the square base.
Area = side × side = $7 \times 7 = 49 \text{ cm}^2$.
* Step 2: Use the volume formula for a pyramid: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$.
$V = \frac{1}{3} \times 49 \times 11$.
* Step 3: Calculate the value.
$49 \times 11 = 539$.
$539 \div 3 = 179.666...$
* Step 4: Round to 1 decimal place.
$179.7 \text{ cm}^3$.
2) Find the total surface area of a cone whose base has a radius of 2.5 cm and slant height is 10 cm. Give your answer in terms of $\pi$.
* Step 1: Identify the formula for Total Surface Area: $A = \pi r^2 + \pi r l$ (Area of base + Area of curved surface).
* Step 2: Plug in the values ($r = 2.5$, $l = 10$).
Base Area = $\pi \times 2.5^2 = 6.25\pi$.
Curved Area = $\pi \times 2.5 \times 10 = 25\pi$.
* Step 3: Add them together.
$6.25\pi + 25\pi = 31.25\pi \text{ cm}^2$.
3) Calculate the volume of a sphere with a diameter of 40 mm.
* Step 1: Find the radius. Radius is half the diameter.
$r = 40 \div 2 = 20 \text{ mm}$.
* Step 2: Use the volume formula for a sphere: $V = \frac{4}{3}\pi r^3$.
$V = \frac{4}{3} \times \pi \times 20^3$.
* Step 3: Calculate $20^3$.
$20 \times 20 \times 20 = 8000$.
* Step 4: Solve for Volume.
$V = \frac{4}{3} \times \pi \times 8000$.
$V \approx 1.3333 \times 3.14159 \times 8000$.
$V \approx 33510.32...$
* Step 5: Round to 1 decimal place.
$33510.3 \text{ mm}^3$.
4) A sphere fits exactly into a cube with sides 18 mm. Calculate the surface area of the sphere. Given your answer in squared centimetres.
* Step 1: Determine the diameter and radius. Since it fits exactly, the diameter equals the side of the cube (18 mm).
Radius ($r$) = $18 \div 2 = 9 \text{ mm}$.
* Step 2: Convert units. The question asks for the answer in cm². It is easier to convert the radius first.
$9 \text{ mm} = 0.9 \text{ cm}$.
* Step 3: Use the surface area formula for a sphere: $A = 4\pi r^2$.
$A = 4 \times \pi \times 0.9^2$.
* Step 4: Calculate.
$0.9^2 = 0.81$.
$4 \times 0.81 = 3.24$.
$3.24 \times \pi \approx 10.1787...$
* Step 5: Round to 1 decimal place.
$10.2 \text{ cm}^2$.
5) A metal sphere of diameter 14 cm is melted down and recast into a solid cylinder of diameter 4 cm. Work out the height of the cylinder.
* Step 1: Find the volume of the sphere.
Radius of sphere ($r_s$) = $14 \div 2 = 7 \text{ cm}$.
Volume = $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi (7^3) = \frac{4}{3}\pi (343) = \frac{1372}{3}\pi \text{ cm}^3$.
* Step 2: Set up the volume of the cylinder.
Radius of cylinder ($r_c$) = $4 \div 2 = 2 \text{ cm}$.
Volume = $\pi r^2 h = \pi (2^2) h = 4\pi h$.
* Step 3: Equate the volumes (since the metal is melted and recast, volume stays the same).
$4\pi h = \frac{1372}{3}\pi$.
* Step 4: Solve for $h$. Divide both sides by $\pi$ and then by 4.
$4h = \frac{1372}{3}$.
$h = \frac{1372}{3 \times 4} = \frac{1372}{12}$.
$h = 114.333...$
* Step 5: Round to 1 decimal place.
$114.3 \text{ cm}$.
6) What is the volume of a cone whose curved surface area is $256\pi \text{ cm}^2$ and slant height 32 cm?
* Step 1: Find the radius using the curved surface area formula ($A = \pi r l$).
$256\pi = \pi \times r \times 32$.
Divide by $\pi$: $256 = 32r$.
$r = 256 \div 32 = 8 \text{ cm}$.
* Step 2: Find the vertical height ($h$) using Pythagoras' theorem ($r^2 + h^2 = l^2$).
$8^2 + h^2 = 32^2$.
$64 + h^2 = 1024$.
$h^2 = 1024 - 64 = 960$.
$h = \sqrt{960} \approx 30.9838... \text{ cm}$.
* Step 3: Calculate the volume ($V = \frac{1}{3}\pi r^2 h$).
$V = \frac{1}{3} \times \pi \times 8^2 \times \sqrt{960}$.
$V = \frac{1}{3} \times \pi \times 64 \times 30.9838...$
$V \approx 67.02 \times 30.9838...$
$V \approx 2076.58...$
* Step 4: Round to 1 decimal place.
$2076.6 \text{ cm}^3$.
Final Answer:
1) 179.7 cm³
2) 31.25π cm²
3) 33510.3 mm³
4) 10.2 cm²
5) 114.3 cm
6) 2076.6 cm³
1) Calculate the volume of a pyramid with a square base of sides 7 cm and a vertical height of 11 cm.
* Step 1: Find the area of the square base.
Area = side × side = $7 \times 7 = 49 \text{ cm}^2$.
* Step 2: Use the volume formula for a pyramid: $V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$.
$V = \frac{1}{3} \times 49 \times 11$.
* Step 3: Calculate the value.
$49 \times 11 = 539$.
$539 \div 3 = 179.666...$
* Step 4: Round to 1 decimal place.
$179.7 \text{ cm}^3$.
2) Find the total surface area of a cone whose base has a radius of 2.5 cm and slant height is 10 cm. Give your answer in terms of $\pi$.
* Step 1: Identify the formula for Total Surface Area: $A = \pi r^2 + \pi r l$ (Area of base + Area of curved surface).
* Step 2: Plug in the values ($r = 2.5$, $l = 10$).
Base Area = $\pi \times 2.5^2 = 6.25\pi$.
Curved Area = $\pi \times 2.5 \times 10 = 25\pi$.
* Step 3: Add them together.
$6.25\pi + 25\pi = 31.25\pi \text{ cm}^2$.
3) Calculate the volume of a sphere with a diameter of 40 mm.
* Step 1: Find the radius. Radius is half the diameter.
$r = 40 \div 2 = 20 \text{ mm}$.
* Step 2: Use the volume formula for a sphere: $V = \frac{4}{3}\pi r^3$.
$V = \frac{4}{3} \times \pi \times 20^3$.
* Step 3: Calculate $20^3$.
$20 \times 20 \times 20 = 8000$.
* Step 4: Solve for Volume.
$V = \frac{4}{3} \times \pi \times 8000$.
$V \approx 1.3333 \times 3.14159 \times 8000$.
$V \approx 33510.32...$
* Step 5: Round to 1 decimal place.
$33510.3 \text{ mm}^3$.
4) A sphere fits exactly into a cube with sides 18 mm. Calculate the surface area of the sphere. Given your answer in squared centimetres.
* Step 1: Determine the diameter and radius. Since it fits exactly, the diameter equals the side of the cube (18 mm).
Radius ($r$) = $18 \div 2 = 9 \text{ mm}$.
* Step 2: Convert units. The question asks for the answer in cm². It is easier to convert the radius first.
$9 \text{ mm} = 0.9 \text{ cm}$.
* Step 3: Use the surface area formula for a sphere: $A = 4\pi r^2$.
$A = 4 \times \pi \times 0.9^2$.
* Step 4: Calculate.
$0.9^2 = 0.81$.
$4 \times 0.81 = 3.24$.
$3.24 \times \pi \approx 10.1787...$
* Step 5: Round to 1 decimal place.
$10.2 \text{ cm}^2$.
5) A metal sphere of diameter 14 cm is melted down and recast into a solid cylinder of diameter 4 cm. Work out the height of the cylinder.
* Step 1: Find the volume of the sphere.
Radius of sphere ($r_s$) = $14 \div 2 = 7 \text{ cm}$.
Volume = $\frac{4}{3}\pi r^3 = \frac{4}{3}\pi (7^3) = \frac{4}{3}\pi (343) = \frac{1372}{3}\pi \text{ cm}^3$.
* Step 2: Set up the volume of the cylinder.
Radius of cylinder ($r_c$) = $4 \div 2 = 2 \text{ cm}$.
Volume = $\pi r^2 h = \pi (2^2) h = 4\pi h$.
* Step 3: Equate the volumes (since the metal is melted and recast, volume stays the same).
$4\pi h = \frac{1372}{3}\pi$.
* Step 4: Solve for $h$. Divide both sides by $\pi$ and then by 4.
$4h = \frac{1372}{3}$.
$h = \frac{1372}{3 \times 4} = \frac{1372}{12}$.
$h = 114.333...$
* Step 5: Round to 1 decimal place.
$114.3 \text{ cm}$.
6) What is the volume of a cone whose curved surface area is $256\pi \text{ cm}^2$ and slant height 32 cm?
* Step 1: Find the radius using the curved surface area formula ($A = \pi r l$).
$256\pi = \pi \times r \times 32$.
Divide by $\pi$: $256 = 32r$.
$r = 256 \div 32 = 8 \text{ cm}$.
* Step 2: Find the vertical height ($h$) using Pythagoras' theorem ($r^2 + h^2 = l^2$).
$8^2 + h^2 = 32^2$.
$64 + h^2 = 1024$.
$h^2 = 1024 - 64 = 960$.
$h = \sqrt{960} \approx 30.9838... \text{ cm}$.
* Step 3: Calculate the volume ($V = \frac{1}{3}\pi r^2 h$).
$V = \frac{1}{3} \times \pi \times 8^2 \times \sqrt{960}$.
$V = \frac{1}{3} \times \pi \times 64 \times 30.9838...$
$V \approx 67.02 \times 30.9838...$
$V \approx 2076.58...$
* Step 4: Round to 1 decimal place.
$2076.6 \text{ cm}^3$.
Final Answer:
1) 179.7 cm³
2) 31.25π cm²
3) 33510.3 mm³
4) 10.2 cm²
5) 114.3 cm
6) 2076.6 cm³
Parent Tip: Review the logic above to help your child master the concept of finding surface area volume worksheet.
