Math worksheet for calculating volume and surface area of cones and spheres.
Worksheet titled "Volume and Surface Area of Cones and Spheres" with eight problems involving calculations of volume, surface area, and radius for cones and spheres, featuring diagrams and measurement labels.
JPG
1811×2560
358.1 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #334892
⭐
Show Answer Key & Explanations
Step-by-step solution for: Volume and Surface Area of Cones and Spheres Worksheet | PDF ...
▼
Show Answer Key & Explanations
Step-by-step solution for: Volume and Surface Area of Cones and Spheres Worksheet | PDF ...
Let’s solve each problem step by step. We’ll use the formulas for spheres and cones:
Sphere:
- Volume = (4/3)πr³
- Surface Area = 4πr²
Cone:
- Volume = (1/3)πr²h
- Curved Surface Area = πrl (where l is slant height)
- Total Surface Area = πr(l + r)
We’ll use π ≈ 3.1416, and round answers to 2 decimal places where needed.
---
Volume = (4/3) × π × (2)³
= (4/3) × π × 8
= (32/3) × π
≈ 10.6667 × 3.1416 ≈ 33.51 cm³
Surface Area = 4 × π × (2)²
= 4 × π × 4
= 16π ≈ 16 × 3.1416 ≈ 50.27 cm²
---
Volume = (4/3) × π × (7)³
= (4/3) × π × 343
= (1372/3) × π ≈ 457.333 × 3.1416 ≈ 1436.76 mm³
Surface Area = 4 × π × (7)²
= 4 × π × 49
= 196π ≈ 196 × 3.1416 ≈ 615.75 mm²
---
Volume = (4/3)πr³ = 180
→ r³ = 180 × 3 / (4π) = 540 / (4π) = 135 / π
→ r³ ≈ 135 / 3.1416 ≈ 42.971
→ r ≈ ∛42.971 ≈ 3.50 cm (rounded to 2 decimals)
---
Surface Area = 4πr² = 25
→ r² = 25 / (4π) ≈ 25 / 12.5664 ≈ 1.9894
→ r ≈ √1.9894 ≈ 1.41 mm
---
Curved Surface Area = πrl = π × 5 × 13 = 65π ≈ 65 × 3.1416 ≈ 204.20 cm²
Total Surface Area = πr(l + r) = π × 5 × (13 + 5) = π × 5 × 18 = 90π ≈ 90 × 3.1416 ≈ 282.74 cm²
*(Note: You can also add base area πr² = 25π to curved surface area 65π → total 90π — same result.)*
---
Wait — we’re given radius and slant height, but not vertical height. To find volume, we need height.
Use Pythagoras: h² + r² = l²
→ h² + 7² = 25²
→ h² + 49 = 625
→ h² = 576 → h = 24 cm
Now Volume = (1/3)πr²h = (1/3) × π × 49 × 24
= (1/3) × 1176 × π = 392π ≈ 392 × 3.1416 ≈ 1231.51 cm³
Total Surface Area = πr(l + r) = π × 7 × (25 + 7) = π × 7 × 32 = 224π ≈ 224 × 3.1416 ≈ 703.72 cm²
---
First, find slant height l using Pythagoras:
l² = r² + h² = (7.2)² + (9.6)² = 51.84 + 92.16 = 144 → l = 12 cm
Volume = (1/3)πr²h = (1/3) × π × (7.2)² × 9.6
= (1/3) × π × 51.84 × 9.6
= (1/3) × 497.664 × π = 165.888π ≈ 165.888 × 3.1416 ≈ 521.15 cm³
Total Surface Area = πr(l + r) = π × 7.2 × (12 + 7.2) = π × 7.2 × 19.2
= 138.24π ≈ 138.24 × 3.1416 ≈ 434.29 cm²
*(Note: Since it’s an inverted cone open at top? But question says “total surface area” — so we include base. The diagram shows a solid cone, so yes, include circular base.)*
---
Given:
- Cone height = 15 cm
- Hemisphere radius = ? Wait — look at diagram: total height from tip to bottom of hemisphere is 23 cm. Hemisphere height = radius. So if cone height is 15 cm, then hemisphere radius = 23 - 15 = 8 cm.
So radius r = 8 cm for both cone and hemisphere.
Volume of whole shape = Volume of cone + Volume of hemisphere
Volume of cone = (1/3)πr²h = (1/3) × π × 64 × 15 = (1/3) × 960 × π = 320π
Volume of hemisphere = (2/3)πr³ = (2/3) × π × 512 = (1024/3)π ≈ 341.333π
Total Volume = 320π + 341.333π = 661.333π ≈ 661.333 × 3.1416 ≈ 2077.58 cm³
Total area = Curved surface area of cone + Curved surface area of hemisphere
*(Note: Base of cone is attached to hemisphere, so not exposed. Hemisphere has no flat base exposed either — it’s sitting on something? Diagram doesn’t show base, so assume only curved surfaces are counted. Also, “Total area” likely means external surface area.)*
Curved SA of cone = πrl → need slant height l.
l = √(r² + h²) = √(64 + 225) = √289 = 17 cm
So curved SA cone = π × 8 × 17 = 136π
Curved SA hemisphere = 2πr² = 2 × π × 64 = 128π
Total Area = 136π + 128π = 264π ≈ 264 × 3.1416 ≈ 829.38 cm²
---
Final Answer:
1) Volume = 33.51 cm³, Surface area = 50.27 cm²
2) Volume = 1436.76 mm³, Surface area = 615.75 mm²
3) Radius = 3.50 cm
4) Radius = 1.41 mm
5) Curved surface area = 204.20 cm², Total surface area = 282.74 cm²
6) Volume = 1231.51 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.29 cm²
8) Volume = 2077.58 cm³, Total area = 829.38 cm²
Sphere:
- Volume = (4/3)πr³
- Surface Area = 4πr²
Cone:
- Volume = (1/3)πr²h
- Curved Surface Area = πrl (where l is slant height)
- Total Surface Area = πr(l + r)
We’ll use π ≈ 3.1416, and round answers to 2 decimal places where needed.
---
Problem 1: Sphere with radius 2 cm
Volume = (4/3) × π × (2)³
= (4/3) × π × 8
= (32/3) × π
≈ 10.6667 × 3.1416 ≈ 33.51 cm³
Surface Area = 4 × π × (2)²
= 4 × π × 4
= 16π ≈ 16 × 3.1416 ≈ 50.27 cm²
---
Problem 2: Sphere with diameter 14 mm → radius = 7 mm
Volume = (4/3) × π × (7)³
= (4/3) × π × 343
= (1372/3) × π ≈ 457.333 × 3.1416 ≈ 1436.76 mm³
Surface Area = 4 × π × (7)²
= 4 × π × 49
= 196π ≈ 196 × 3.1416 ≈ 615.75 mm²
---
Problem 3: Sphere with volume 180 cm³ → find radius
Volume = (4/3)πr³ = 180
→ r³ = 180 × 3 / (4π) = 540 / (4π) = 135 / π
→ r³ ≈ 135 / 3.1416 ≈ 42.971
→ r ≈ ∛42.971 ≈ 3.50 cm (rounded to 2 decimals)
---
Problem 4: Sphere with surface area 25 mm² → find radius
Surface Area = 4πr² = 25
→ r² = 25 / (4π) ≈ 25 / 12.5664 ≈ 1.9894
→ r ≈ √1.9894 ≈ 1.41 mm
---
Problem 5: Cone with radius 5 cm, height 12 cm, slant height 13 cm
Curved Surface Area = πrl = π × 5 × 13 = 65π ≈ 65 × 3.1416 ≈ 204.20 cm²
Total Surface Area = πr(l + r) = π × 5 × (13 + 5) = π × 5 × 18 = 90π ≈ 90 × 3.1416 ≈ 282.74 cm²
*(Note: You can also add base area πr² = 25π to curved surface area 65π → total 90π — same result.)*
---
Problem 6: Cone with radius 7 cm, slant height 25 cm → need height first?
Wait — we’re given radius and slant height, but not vertical height. To find volume, we need height.
Use Pythagoras: h² + r² = l²
→ h² + 7² = 25²
→ h² + 49 = 625
→ h² = 576 → h = 24 cm
Now Volume = (1/3)πr²h = (1/3) × π × 49 × 24
= (1/3) × 1176 × π = 392π ≈ 392 × 3.1416 ≈ 1231.51 cm³
Total Surface Area = πr(l + r) = π × 7 × (25 + 7) = π × 7 × 32 = 224π ≈ 224 × 3.1416 ≈ 703.72 cm²
---
Problem 7: Inverted cone with diameter 14.4 cm → radius = 7.2 cm, height = 9.6 cm
First, find slant height l using Pythagoras:
l² = r² + h² = (7.2)² + (9.6)² = 51.84 + 92.16 = 144 → l = 12 cm
Volume = (1/3)πr²h = (1/3) × π × (7.2)² × 9.6
= (1/3) × π × 51.84 × 9.6
= (1/3) × 497.664 × π = 165.888π ≈ 165.888 × 3.1416 ≈ 521.15 cm³
Total Surface Area = πr(l + r) = π × 7.2 × (12 + 7.2) = π × 7.2 × 19.2
= 138.24π ≈ 138.24 × 3.1416 ≈ 434.29 cm²
*(Note: Since it’s an inverted cone open at top? But question says “total surface area” — so we include base. The diagram shows a solid cone, so yes, include circular base.)*
---
Problem 8: Composite shape — cone on top of hemisphere. Both share same radius.
Given:
- Cone height = 15 cm
- Hemisphere radius = ? Wait — look at diagram: total height from tip to bottom of hemisphere is 23 cm. Hemisphere height = radius. So if cone height is 15 cm, then hemisphere radius = 23 - 15 = 8 cm.
So radius r = 8 cm for both cone and hemisphere.
Volume of whole shape = Volume of cone + Volume of hemisphere
Volume of cone = (1/3)πr²h = (1/3) × π × 64 × 15 = (1/3) × 960 × π = 320π
Volume of hemisphere = (2/3)πr³ = (2/3) × π × 512 = (1024/3)π ≈ 341.333π
Total Volume = 320π + 341.333π = 661.333π ≈ 661.333 × 3.1416 ≈ 2077.58 cm³
Total area = Curved surface area of cone + Curved surface area of hemisphere
*(Note: Base of cone is attached to hemisphere, so not exposed. Hemisphere has no flat base exposed either — it’s sitting on something? Diagram doesn’t show base, so assume only curved surfaces are counted. Also, “Total area” likely means external surface area.)*
Curved SA of cone = πrl → need slant height l.
l = √(r² + h²) = √(64 + 225) = √289 = 17 cm
So curved SA cone = π × 8 × 17 = 136π
Curved SA hemisphere = 2πr² = 2 × π × 64 = 128π
Total Area = 136π + 128π = 264π ≈ 264 × 3.1416 ≈ 829.38 cm²
---
Final Answer:
1) Volume = 33.51 cm³, Surface area = 50.27 cm²
2) Volume = 1436.76 mm³, Surface area = 615.75 mm²
3) Radius = 3.50 cm
4) Radius = 1.41 mm
5) Curved surface area = 204.20 cm², Total surface area = 282.74 cm²
6) Volume = 1231.51 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.29 cm²
8) Volume = 2077.58 cm³, Total area = 829.38 cm²
Parent Tip: Review the logic above to help your child master the concept of volume of a cylinder cone and sphere worksheet.
