Math worksheet for calculating the surface area of cylinders and cones.
Worksheet titled "Surface Area of Cylinders and Cones" with nine problems, each showing a cylinder or cone with dimensions and a space to calculate the surface area.
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Step-by-step solution for: Geometry Worksheets | Volume Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Volume Worksheets
To find the surface area of cylinders and cones, we use specific formulas. Let's solve each problem step by step.
Formulas:
* Cylinder Surface Area: $SA = 2\pi r^2 + 2\pi rh$ (Area of two circles + Area of the curved side)
* Cone Surface Area: $SA = \pi r^2 + \pi rl$ (Area of the base circle + Area of the curved side), where $l$ is the slant height.
* Use $\pi \approx 3.14159$ for calculations. Round to the nearest hundredth.
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1) Cylinder: radius = 8 ft, height = 12 ft
* Base Area ($2\pi r^2$): $2 \times \pi \times 8^2 = 128\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 8 \times 12 = 192\pi$
* Total SA: $128\pi + 192\pi = 320\pi$
* Calculation: $320 \times 3.14159 \approx 1005.31$
* Answer: 1,005.31 ft²
2) Cone: radius = 4 in, slant height = 6 in
* Base Area ($\pi r^2$): $\pi \times 4^2 = 16\pi$
* Lateral Area ($\pi rl$): $\pi \times 4 \times 6 = 24\pi$
* Total SA: $16\pi + 24\pi = 40\pi$
* Calculation: $40 \times 3.14159 \approx 125.66$
* Answer: 125.66 in²
3) Cylinder: radius = 6 yd, height = 8 yd
* Base Area ($2\pi r^2$): $2 \times \pi \times 6^2 = 72\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 6 \times 8 = 96\pi$
* Total SA: $72\pi + 96\pi = 168\pi$
* Calculation: $168 \times 3.14159 \approx 527.79$
* Answer: 527.79 yd²
4) Cone: radius = 5 mm, slant height = 13 mm
* Base Area ($\pi r^2$): $\pi \times 5^2 = 25\pi$
* Lateral Area ($\pi rl$): $\pi \times 5 \times 13 = 65\pi$
* Total SA: $25\pi + 65\pi = 90\pi$
* Calculation: $90 \times 3.14159 \approx 282.74$
* Answer: 282.74 mm²
5) Cone: radius = 6 mm, slant height = 14 mm
* Base Area ($\pi r^2$): $\pi \times 6^2 = 36\pi$
* Lateral Area ($\pi rl$): $\pi \times 6 \times 14 = 84\pi$
* Total SA: $36\pi + 84\pi = 120\pi$
* Calculation: $120 \times 3.14159 \approx 376.99$
* Answer: 376.99 mm²
6) Cylinder: radius = 6 cm, height = 8 cm
* Base Area ($2\pi r^2$): $2 \times \pi \times 6^2 = 72\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 6 \times 8 = 96\pi$
* Total SA: $72\pi + 96\pi = 168\pi$
* Calculation: $168 \times 3.14159 \approx 527.79$
* Answer: 527.79 cm²
7) Cylinder: radius = 7 yd, height = 10 yd
* Base Area ($2\pi r^2$): $2 \times \pi \times 7^2 = 98\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 7 \times 10 = 140\pi$
* Total SA: $98\pi + 140\pi = 238\pi$
* Calculation: $238 \times 3.14159 \approx 747.70$
* Answer: 747.70 yd²
8) Cone: radius = 7 ft, slant height = 10 ft
* Base Area ($\pi r^2$): $\pi \times 7^2 = 49\pi$
* Lateral Area ($\pi rl$): $\pi \times 7 \times 10 = 70\pi$
* Total SA: $49\pi + 70\pi = 119\pi$
* Calculation: $119 \times 3.14159 \approx 373.85$
* Answer: 373.85 ft²
9) Cone: radius = 4 cm, slant height = 6 cm
* Base Area ($\pi r^2$): $\pi \times 4^2 = 16\pi$
* Lateral Area ($\pi rl$): $\pi \times 4 \times 6 = 24\pi$
* Total SA: $16\pi + 24\pi = 40\pi$
* Calculation: $40 \times 3.14159 \approx 125.66$
* Answer: 125.66 cm²
──────────────────────────────────────
Final Answer:
1) 1,005.31 ft²
2) 125.66 in²
3) 527.79 yd²
4) 282.74 mm²
5) 376.99 mm²
6) 527.79 cm²
7) 747.70 yd²
8) 373.85 ft²
9) 125.66 cm²
Formulas:
* Cylinder Surface Area: $SA = 2\pi r^2 + 2\pi rh$ (Area of two circles + Area of the curved side)
* Cone Surface Area: $SA = \pi r^2 + \pi rl$ (Area of the base circle + Area of the curved side), where $l$ is the slant height.
* Use $\pi \approx 3.14159$ for calculations. Round to the nearest hundredth.
---
Step-by-Step Solutions
1) Cylinder: radius = 8 ft, height = 12 ft
* Base Area ($2\pi r^2$): $2 \times \pi \times 8^2 = 128\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 8 \times 12 = 192\pi$
* Total SA: $128\pi + 192\pi = 320\pi$
* Calculation: $320 \times 3.14159 \approx 1005.31$
* Answer: 1,005.31 ft²
2) Cone: radius = 4 in, slant height = 6 in
* Base Area ($\pi r^2$): $\pi \times 4^2 = 16\pi$
* Lateral Area ($\pi rl$): $\pi \times 4 \times 6 = 24\pi$
* Total SA: $16\pi + 24\pi = 40\pi$
* Calculation: $40 \times 3.14159 \approx 125.66$
* Answer: 125.66 in²
3) Cylinder: radius = 6 yd, height = 8 yd
* Base Area ($2\pi r^2$): $2 \times \pi \times 6^2 = 72\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 6 \times 8 = 96\pi$
* Total SA: $72\pi + 96\pi = 168\pi$
* Calculation: $168 \times 3.14159 \approx 527.79$
* Answer: 527.79 yd²
4) Cone: radius = 5 mm, slant height = 13 mm
* Base Area ($\pi r^2$): $\pi \times 5^2 = 25\pi$
* Lateral Area ($\pi rl$): $\pi \times 5 \times 13 = 65\pi$
* Total SA: $25\pi + 65\pi = 90\pi$
* Calculation: $90 \times 3.14159 \approx 282.74$
* Answer: 282.74 mm²
5) Cone: radius = 6 mm, slant height = 14 mm
* Base Area ($\pi r^2$): $\pi \times 6^2 = 36\pi$
* Lateral Area ($\pi rl$): $\pi \times 6 \times 14 = 84\pi$
* Total SA: $36\pi + 84\pi = 120\pi$
* Calculation: $120 \times 3.14159 \approx 376.99$
* Answer: 376.99 mm²
6) Cylinder: radius = 6 cm, height = 8 cm
* Base Area ($2\pi r^2$): $2 \times \pi \times 6^2 = 72\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 6 \times 8 = 96\pi$
* Total SA: $72\pi + 96\pi = 168\pi$
* Calculation: $168 \times 3.14159 \approx 527.79$
* Answer: 527.79 cm²
7) Cylinder: radius = 7 yd, height = 10 yd
* Base Area ($2\pi r^2$): $2 \times \pi \times 7^2 = 98\pi$
* Lateral Area ($2\pi rh$): $2 \times \pi \times 7 \times 10 = 140\pi$
* Total SA: $98\pi + 140\pi = 238\pi$
* Calculation: $238 \times 3.14159 \approx 747.70$
* Answer: 747.70 yd²
8) Cone: radius = 7 ft, slant height = 10 ft
* Base Area ($\pi r^2$): $\pi \times 7^2 = 49\pi$
* Lateral Area ($\pi rl$): $\pi \times 7 \times 10 = 70\pi$
* Total SA: $49\pi + 70\pi = 119\pi$
* Calculation: $119 \times 3.14159 \approx 373.85$
* Answer: 373.85 ft²
9) Cone: radius = 4 cm, slant height = 6 cm
* Base Area ($\pi r^2$): $\pi \times 4^2 = 16\pi$
* Lateral Area ($\pi rl$): $\pi \times 4 \times 6 = 24\pi$
* Total SA: $16\pi + 24\pi = 40\pi$
* Calculation: $40 \times 3.14159 \approx 125.66$
* Answer: 125.66 cm²
──────────────────────────────────────
Final Answer:
1) 1,005.31 ft²
2) 125.66 in²
3) 527.79 yd²
4) 282.74 mm²
5) 376.99 mm²
6) 527.79 cm²
7) 747.70 yd²
8) 373.85 ft²
9) 125.66 cm²
Parent Tip: Review the logic above to help your child master the concept of surface area word problems worksheet with answers.
