Square Pyramid Practice Problems

Formulas you need

• Volume (any pyramid): $V = \tfrac{1}{3} B h$
• Slant height: $\ell = \sqrt{(b/2)^2 + h^2}$
• Volume (square base): $V = \tfrac{1}{3} b^2 h$
• Lateral surface: $LSA = 2 b \ell$
• Surface area: $SA = b^2 + 2 b \ell$

where B = area of the base (any pyramid), b = base edge length (square base), h = vertical height, ℓ = slant height (face), V = volume

Generate a new problem

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 square pyramid has base edge b = 5 and vertical height h = 2. Find the slant height (ℓ).

   Answer & working

   3.2 units

   slant ℓ = √((b/2)² + h²) = √(6.25 + 4) ≈ 3.2

2. A square pyramid has base edge b = 5 and vertical height h = 5. Find the volume (V).

   Answer & working

   41.67 units³

   V = ⅓ · b² · h = ⅓ · 5² · 5 ≈ 41.67

3. A square pyramid has base edge b = 1 and vertical height h = 6. Find the lateral surface (LSA).

   Answer & working

   12.04 units²

   LSA = 2 · b · ℓ = 2 · 1 · 6.0208 ≈ 12.04

4. A square pyramid has base edge b = 4 and vertical height h = 2. Find the surface area (SA).

   Answer & working

   38.63 units²

   SA = b² + 2bℓ = 16 + 22.6274 ≈ 38.63

5. A square pyramid has base edge b = 2 and vertical height h = 1. Find the slant height (ℓ).

   Answer & working

   1.41 units

   slant ℓ = √((b/2)² + h²) = √(1 + 1) ≈ 1.41

Core practice

1. A square pyramid has base edge b = 10 and vertical height h = 9. Find the slant height (ℓ).

   Answer & working

   10.3 units

   slant ℓ = √((b/2)² + h²) = √(25 + 81) ≈ 10.3

2. A square pyramid has base edge b = 1 and vertical height h = 5. Find the volume (V).

   Answer & working

   1.67 units³

   V = ⅓ · b² · h = ⅓ · 1² · 5 ≈ 1.67

3. A square pyramid has base edge b = 10 and vertical height h = 12. Find the lateral surface (LSA).

   Answer & working

   260 units²

   LSA = 2 · b · ℓ = 2 · 10 · 13 = 260

4. A square pyramid has base edge b = 7 and vertical height h = 12. Find the surface area (SA).

   Answer & working

   224 units²

   SA = b² + 2bℓ = 49 + 175 = 224

5. A square pyramid has base edge b = 1 and vertical height h = 1. Find the slant height (ℓ).

   Answer & working

   1.12 units

   slant ℓ = √((b/2)² + h²) = √(0.25 + 1) ≈ 1.12

Stretch

1. A square pyramid has base edge b = 8.5 and vertical height h = 2.5. Find the slant height (ℓ).

   Answer & working

   4.93 units

   slant ℓ = √((b/2)² + h²) = √(18.0625 + 6.25) ≈ 4.93

2. A square pyramid has base edge b = 5.5 and vertical height h = 1.5. Find the volume (V).

   Answer & working

   15.12 units³

   V = ⅓ · b² · h = ⅓ · 5.5² · 1.5 ≈ 15.12

3. A square pyramid has base edge b = 2.5 and vertical height h = 2.5. Find the lateral surface (LSA).

   Answer & working

   13.98 units²

   LSA = 2 · b · ℓ = 2 · 2.5 · 2.7951 ≈ 13.98

4. A square pyramid has base edge b = 4.5 and vertical height h = 10.5. Find the surface area (SA).

   Answer & working

   116.9 units²

   SA = b² + 2bℓ = 20.25 + 96.6453 ≈ 116.9

5. A square pyramid has base edge b = 6.5 and vertical height h = 2.5. Find the slant height (ℓ).

   Answer & working

   4.1 units

   slant ℓ = √((b/2)² + h²) = √(10.5625 + 6.25) ≈ 4.1