Cone Practice Problems

Formulas you need

• Slant height: $\ell = \sqrt{r^2 + h^2}$
• Volume: $V = \tfrac{1}{3}\pi r^2 h$
• Lateral surface: $LSA = \pi r \ell$
• Surface area: $SA = \pi r(r + \ell)$

where r = base radius, h = vertical height, ℓ = slant height, V = volume

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Pick a difficulty, work it out, then check your answer. Every press gives a new question — the numbers are drawn fresh each time, so you can keep going.

Warm-up

1. A cone has radius r = 4 and vertical height h = 2. Find the slant height (ℓ).

   Answer & working

   4.47 units

   slant ℓ = √(r² + h²) = √(16 + 4) ≈ 4.47

2. A cone has radius r = 1 and vertical height h = 5. Find the volume (V).

   Answer & working

   5.24 units³

   V = ⅓πr²h = ⅓ · π · 1² · 5 = 5π/3 ≈ 5.24

3. A cone has radius r = 4 and vertical height h = 5. Find the lateral surface (LSA).

   Answer & working

   80.46 units²

   LSA = πrℓ = π · 4 · 6.4031 = 25.6125π ≈ 80.46

4. A cone has radius r = 4 and vertical height h = 5. Find the surface area (SA).

   Answer & working

   130.73 units²

   SA = πr(r + ℓ) = π · 4(4 + 6.4031) = 41.6125π ≈ 130.73

5. A cone has radius r = 1 and vertical height h = 3. Find the slant height (ℓ).

   Answer & working

   3.16 units

   slant ℓ = √(r² + h²) = √(1 + 9) ≈ 3.16

Core practice

1. A cone has radius r = 3 and vertical height h = 8. Find the slant height (ℓ).

   Answer & working

   8.54 units

   slant ℓ = √(r² + h²) = √(9 + 64) ≈ 8.54

2. A cone has radius r = 2 and vertical height h = 5. Find the volume (V).

   Answer & working

   20.94 units³

   V = ⅓πr²h = ⅓ · π · 2² · 5 = 20π/3 ≈ 20.94

3. A cone has radius r = 7 and vertical height h = 11. Find the lateral surface (LSA).

   Answer & working

   286.73 units²

   LSA = πrℓ = π · 7 · 13.0384 = 91.2688π ≈ 286.73

4. A cone has radius r = 3 and vertical height h = 6. Find the surface area (SA).

   Answer & working

   91.5 units²

   SA = πr(r + ℓ) = π · 3(3 + 6.7082) = 29.1246π ≈ 91.5

5. A cone has radius r = 8 and vertical height h = 6. Find the slant height (ℓ).

   Answer & working

   10 units

   slant ℓ = √(r² + h²) = √(64 + 36) = 10

Stretch

1. A cone has radius r = 6.5 and vertical height h = 7.5. Find the slant height (ℓ).

   Answer & working

   9.92 units

   slant ℓ = √(r² + h²) = √(42.25 + 56.25) ≈ 9.92

2. A cone has radius r = 4.5 and vertical height h = 2.5. Find the volume (V).

   Answer & working

   53.01 units³

   V = ⅓πr²h = ⅓ · π · 4.5² · 2.5 = 16.875π ≈ 53.01

3. A cone has radius r = 2.5 and vertical height h = 5.5. Find the lateral surface (LSA).

   Answer & working

   47.45 units²

   LSA = πrℓ = π · 2.5 · 6.0415 = 15.1038π ≈ 47.45

4. A cone has radius r = 2.5 and vertical height h = 12. Find the surface area (SA).

   Answer & working

   115.91 units²

   SA = πr(r + ℓ) = π · 2.5(2.5 + 12.2577) = 36.8941π ≈ 115.91

5. A cone has radius r = 1.5 and vertical height h = 1.5. Find the slant height (ℓ).

   Answer & working

   2.12 units

   slant ℓ = √(r² + h²) = √(2.25 + 2.25) ≈ 2.12