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.

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 — area (a + b) × l / 2 per face, times 4 faces = 2(a + b)l for all lateral surface.

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 (rarely finished — sits under the sculpture). Bottom: 80² = 6,400 cm² (sits on ground — usually not finished either). Four trapezoidal sides: 2 × (50 + 80) × 61.85 = 2 × 130 × 61.85 ≈ 16,081 cm² = 1.61 m² (the VISIBLE finishing surface).

For a stone-clad finish at ~25 kg/m² stone: 40 kg of stone for the four visible sides.

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 measure the slant directly along the trapezoidal face, you can skip the calculation — but for most real measurements, h is the easier dimension to capture.

Top and bottom — included or excluded?

Most real applications care about just the FOUR SIDE FACES. The top sits under whatever’s mounted on the pedestal, the bottom sits on the floor. Exclude both for typical paint/coating estimates:

• 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. ✓