Truncated Pyramid Volume Calculator

Truncated Pyramid (Frustum)

Calculate volume of a truncated pyramid

Formula:

V = (1/3) * h * (A₁ + A₂ + √(A₁ * A₂))

Where A₁ and A₂ are the areas of the top and bottom bases, and h is the height.

How to Calculate Truncated Pyramid Volume

Step 1: Determine Top Base Area

Measure the top base dimensions and calculate its area (A₁).

Why: The top area is one of the three components in the volume formula. It's usually smaller than the base.

Step 2: Determine Bottom Base Area

Measure the bottom base dimensions and calculate its area (A₂).

Why: The bottom base is typically larger. For similar pyramids, ratio A₂/A₁ relates to (R₃) where R is the scaling factor.

Step 3: Measure Height Perpendicular

Measure the perpendicular distance between the two parallel base planes.

Why: Height must be perpendicular to both bases. Slant measurements give incorrect results.

Step 4: Calculate the Geometric Mean

Compute √(A₁ * A₂) as part of the formula.

Why: The geometric mean of the two base areas appears in the formula. It represents the average cross-sectional area through the frustum.

Step 5: Verify and Document Results

Check that all values are positive and the volume makes geometric sense.

Why: Volume should be between that of a prism with area A₁ and one with area A₂. Extreme values indicate errors.

Real-World Example

Truncated Pyramid Storage Tank

Scenario: Storage tank: top 4×4m, bottom 6×6m (square bases), height 5m. What is volume?
Step 1: Top area A₁ = 4 × 4 = 16 m²
Step 2: Bottom area A₂ = 6 × 6 = 36 m²
Step 3: Height h = 5 m (perpendicular)
Step 4: Geometric mean = √(16 × 36) = √576 = 24 m²
Step 5: V = (1/3) × 5 × (16 + 36 + 24) = (1/3) × 5 × 76 = 380/3 ≈ 126.67 m³
Verification: Result between (16×5=80) and (36×5=180) m³ ✓; geometrically sound
Result: Tank capacity is 126.67 cubic meters or approximately 126,670 liters
Interpretation: The tank holds about 126 m³ of liquid. This capacity determines pumping requirements, material needed, and structural loads.

Related Tools