Question

Surface Area of Cylinders Quick Check...I need help

Answer 1

Sure — I can help. Quick summary, formulas, short worked examples, then a few practice problems with answers so you can check yourself.

Key formulas (for a right circular cylinder)
- Radius = r, height = h.
- Lateral (side) surface area = 2πrh. (This is the rectangle you get when you "unroll" the side; its width = circumference = 2πr, height = h.)
- Area of one circular base = πr^2.
- Total surface area (closed cylinder, both top and bottom) = 2πrh + 2πr^2 = 2πr(h + r).
- If the cylinder is open (no top), total area = 2πrh + πr^2 (just one base).
- If diameter d is given: r = d/2. If circumference C is given: r = C/(2π), and lateral area = C × h.

Worked examples
1) r = 3 cm, h = 5 cm
- Lateral = 2πrh = 2π(3)(5) = 30π ≈ 94.25 cm^2
- Total = 2πr(h + r) = 2π(3)(5 + 3) = 48π ≈ 150.80 cm^2

2) d = 10 in, h = 7 in → r = 5 in
- Lateral = 2π(5)(7) = 70π ≈ 219.91 in^2
- Total = 70π + 2π(25) = 120π ≈ 376.99 in^2

3) Given circumference C = 31.4 cm (≈ 10π), h = 4 cm
- r = C/(2π) = 31.4/(2π) ≈ 5 cm
- Lateral = C × h = 31.4 × 4 = 125.6 cm^2
- Total = lateral + 2πr^2 = 125.6 + 2π(25) ≈ 125.6 + 157.08 = 282.68 cm^2

Common tips
- Keep units consistent and attach units (cm^2, in^2, m^2).
- Decide whether the cylinder is closed (2 bases) or open (1 or 0 bases) before adding base areas.
- For answers, say whether you’re leaving in terms of π (e.g., 48π) or giving a decimal (use the same approach consistently).

Practice quick check (answers below)
1) r = 4 cm, h = 10 cm — find lateral and total.
2) d = 8 in, h = 6 in — find total area.
3) C = 18π m, h = 12 m — find lateral and total.
4) Open cylinder (no top), r = 2 cm, h = 7 cm — find total surface area.
5) r = 6 cm, h = 0.5 cm — find lateral and total.

Answers
1) Lateral = 80π ≈ 251.33 cm^2; Total = 112π ≈ 351.86 cm^2.
2) r = 4 in → Total = 80π ≈ 251.33 in^2.
3) r = 9 m → Lateral = 216π ≈ 678.58 m^2; Total = 378π ≈ 1187.66 m^2.
4) Lateral = 28π ≈ 87.96 cm^2; Base = 4π ≈ 12.57 cm^2; Total (open) = 32π ≈ 100.53 cm^2.
5) Lateral = 2π(6)(0.5) = 6π ≈ 18.85 cm^2; Total = 6π + 2π(36) = 78π ≈ 245.04 cm^2.

If you have a specific problem or worksheet image, paste it here and I’ll walk through the steps with you.