Square Pyramid — Volume + Surface Area
Square pyramid: V = ⅓a²h, slant ℓ = √((a/2)² + h²), lateral SA = 2aℓ, total SA = a² + 2aℓ.
Result
Volume
16.0000
⅓a²h = 16.0000. Slant ℓ = 3.6056.
- Base edge (a)4
- Height (h)3
- Slant height (ℓ)3.605551
- Lateral edge4.123106
- Volume (⅓a²h)16.000000
- Base area (a²)16.000000
- Lateral SA (4 × ½ × a × ℓ)28.844410
- Total SA44.844410
Step-by-step
- Slant ℓ = √((a/2)² + h²) = √(4.0000 + 9.0000) = 3.6056.
- V = ⅓ × base × height = ⅓ × 16 × 3 = 16.0000.
- Lateral SA = 4 × (½ × a × ℓ) = 2aℓ = 2 × 4 × 3.6056 = 28.8444.
- Total SA = base + lateral = 16 + 28.8444 = 44.8444.
How to use this calculator
- Enter base edge and perpendicular height (apex above center).
- Read slant, volume, and surface areas.
About this calculator
A right square pyramid has a square base and four congruent isosceles-triangle faces meeting at an apex above the base center. Volume V = ⅓ × base × height = ⅓a²h (the ⅓ factor applies to all pyramids and cones). Slant height (apex-to-mid-edge along a triangular face) ℓ = √((a/2)² + h²). Lateral SA = 4 × (½ × a × ℓ) = 2aℓ. Total SA = base + lateral = a² + 2aℓ. Source: Stewart, "Calculus" §6.2; Larson, "Geometry" Ch. 11. The Great Pyramid of Giza is the canonical example: a ≈ 230 m, h ≈ 147 m.
Frequently asked
Calculus: integrate cross-sections that scale as (z/h)² × a² from 0 to h. Result: ⅓a²h. Three identical pyramids fit inside a prism with the same base and height.
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