Let’s solve each problem step by step.
We’ll use these formulas:
For cones:
- Curved surface area = π × r × l (where r = radius, l = slant height)
- Total surface area = π × r × (r + l) → or curved + base area (πr²)
For spheres:
- Surface area = 4 × π × r²
- If given diameter, divide by 2 to get radius.
Use π ≈ 3.1416 and round answers to 2 decimal places as instructed.
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Problem 1:
Cone with radius = 5 cm, slant height = 10 cm
Curved surface area = π × 5 × 10 = 50π ≈ 157.08 cm²
Total surface area = π × 5 × (5 + 10) = π × 5 × 15 = 75π ≈ 235.62 cm²
✔ Check: Base area = π×5² = 78.54; 157.08 + 78.54 = 235.62 — correct.
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Problem 2:
Same cone? Wait — looks like same dimensions: radius = 5 cm, slant height = 10 cm
But note: it says “curved surface area” and “total surface area” again — probably same as #1? But let’s confirm from image description.
Actually, looking at the layout, problems 1 and 2 appear identical in diagram — both show cone with r=5, l=10. So likely same answer.
But wait — maybe typo? Or perhaps intentional repetition for practice? We’ll compute same values.
Curved surface area = 157.08 cm²
Total surface area = 235.62 cm²
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Problem 3:
Cone with radius = 9 cm, slant height = 10.7 cm
Curved surface area = π × 9 × 10.7 = π × 96.3 ≈ 302.53 cm²
Total surface area = π × 9 × (9 + 10.7) = π × 9 × 19.7 = π × 177.3 ≈ 557.01 cm²
Check: base = π×81 ≈ 254.47; 302.53 + 254.47 = 557.00 — close enough (rounding).
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Problem 4:
Cone with diameter = 7.2 mm → so radius = 3.6 mm, slant height = 7.7 mm
Curved surface area = π × 3.6 × 7.7 = π × 27.72 ≈ 87.09 mm²
Total surface area = π × 3.6 × (3.6 + 7.7) = π × 3.6 × 11.3 = π × 40.68 ≈ 127.80 mm²
Check: base = π×(3.6)² = π×12.96 ≈ 40.72; 87.09 + 40.72 = 127.81 — good.
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Problem 5:
Sphere with diameter = 3.8 m → radius = 1.9 m
Surface area = 4 × π × (1.9)² = 4 × π × 3.61 = 14.44π ≈ 45.36 m²
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Problem 6:
Sphere with radius = 2.2 cm
Surface area = 4 × π × (2.2)² = 4 × π × 4.84 = 19.36π ≈ 60.82 cm²
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Problem 7:
Sphere with diameter = 16.6 cm → radius = 8.3 cm
Surface area = 4 × π × (8.3)² = 4 × π × 68.89 = 275.56π ≈ 865.69 cm²
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Problem 8:
Given surface area = 300 cm² → find radius.
Formula: SA = 4πr² → so r² = SA / (4π) = 300 / (4 × π) = 300 / 12.5664 ≈ 23.873
Then r = √23.873 ≈ 4.89 cm
Check: 4 × π × (4.89)² ≈ 4 × 3.1416 × 23.9121 ≈ 12.5664 × 23.9121 ≈ 300.5 — very close, rounding error acceptable.
More precisely:
r² = 300 / (4π) = 75 / π ≈ 75 / 3.1415926535 ≈ 23.87324146
r = √23.87324146 ≈ 4.886 → rounded to 2 decimals:
4.89 cm
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Now compiling all answers:
1) Curved: 157.08 cm², Total: 235.62 cm²
2) Curved: 157.08 cm², Total: 235.62 cm²
3) Curved: 302.53 cm², Total: 557.01 cm²
4) Curved: 87.09 mm², Total: 127.80 mm²
5) Surface area: 45.36 m²
6) Surface area: 60.82 cm²
7) Surface area: 865.69 cm²
8) Radius: 4.89 cm
Final Answer:
1) 157.08, 235.62
2) 157.08, 235.62
3) 302.53, 557.01
4) 87.09, 127.80
5) 45.36
6) 60.82
7) 865.69
8) 4.89
Parent Tip: Review the logic above to help your child master the concept of geometry surface area and volume worksheet answers.
