Truncated Pyramid Surface Area Calculator

A square truncated pyramid (square frustum) has six faces: a smaller square top, a larger square bottom, and four congruent trapezoidal sides.

This calculator returns the closed total, all six faces. Real plinths and hoppers rarely need all six, so take the variant you want from the breakdown in the result rather than the headline figure.

SA = a² + b² + 2 × (a + b) × l

Where:

• a = top edge (smaller square)
• b = bottom edge (larger square)
• h = vertical height
• l = slant height of each trapezoidal side face

The slant height is computed from h and the half-difference of the squares: l = √(h² + ((b − a) / 2)²)

Each trapezoidal face has parallel sides a and b with slant height l between them, giving (a + b) × l / 2 per face. Four faces make 2(a + b)l of lateral surface.

Note the half-difference in the slant formula. The face leans back by (b − a)/2, not by (b − a), because the taper is shared between the two opposite sides. Using the full difference is the commonest mistake here and it overstates every side face.

Worked example: concrete plinth pedestal A solid plinth for a garden sculpture: top 50 cm × 50 cm, base 80 cm × 80 cm, height 60 cm. a = 50, b = 80, h = 60. Half-difference: (80 − 50) / 2 = 15 cm. Slant: l = √(60² + 15²) = √(3,600 + 225) = √3,825 ≈ 61.85 cm.

Top: 50² = 2,500 cm², which sits under the sculpture and is rarely finished. Bottom: 80² = 6,400 cm², on the ground and usually not finished either. Four trapezoidal sides: 2 × (50 + 80) × 61.85 ≈ 16,081 cm² = 1.61 m², and that is the visible surface.

At the 25 kg per square metre that a thin stone veneer typically runs, the four sides take about 40 kg. Check the figure against the actual product, since veneer weight varies with thickness and stone type.

Worked example: square grain hopper sheet metal A hopper from 4 m × 4 m to 0.5 m × 0.5 m, height 3 m. a = 0.5, b = 4, h = 3. Half-difference: (4 − 0.5) / 2 = 1.75 m. Slant: l = √(9 + 3.0625) = √12.0625 ≈ 3.47 m. Four trapezoid sides: 2 × (0.5 + 4) × 3.47 ≈ 31.27 m² of steel plate.

That’s the lateral steel area for the hopper. Add 5-10% for seam allowances and edge tabs.

Where truncated pyramid surface area matters:

• Hopper sheet metal estimation. Steel or stainless plate for square hopper bottoms.
• Plinth and pedestal stone cladding. Decorative stone or veneer for sculpture mounts.
• Mesoamerican temple restoration. Stone coverage for stepped truncated-pyramid structures.
• Concrete formwork for pouring truncated-pyramid foundations.
• Modern architectural truncated-pyramid forms in convention centers and theaters.

The slant height calculation:

A common mistake: using h directly as the slant height. For a truncated pyramid with non-trivial taper (a ≠ b), the slant height l is ALWAYS larger than h.

For (b − a) « b: l ≈ h (the taper is small, slant nearly vertical). For (b − a) ≈ b/2 (significant taper): l = √(h² + (b/4)²), noticeably longer than h.

If you can lay a tape along the face itself, use that and skip the arithmetic. On most real objects the vertical height is the easier dimension to capture, which is why the formula starts there.

Top and bottom, included or excluded?

Most real applications want just the FOUR SIDE FACES. The top sits under whatever is mounted on the pedestal and the bottom sits on the floor, so neither gets finished:

• Lateral only: SA_lat = 2(a + b) × l
• With top: + a²
• With bottom: + b²

For a fully exposed truncated pyramid (decorative, freestanding): include all six faces. For a typical plinth or hopper: usually lateral only.

Sanity check:

• a = b: l = h. SA = 2b² + 4bh. Matches square prism formula (with both ends and 4 sides). ✓
• a = 0: l = √(h² + (b/2)²). SA = b² + 2b × l. Matches square pyramid formula (base + 4 triangles). ✓
• h = 0: l = (b − a)/2. SA = a² + b² + (a + b)(b − a) = a² + b² + b² − a² = 2b². Two stacked squares; degenerate. ✓