Square Pyramid Volume Calculator

Enter a square pyramid's base edge length and height to get its volume (V = ⅓s²h), slant height, and surface area.

Quick Facts

Volume formula
V = ⅓ × s² × h
One-third of base area times the vertical height.
Slant height
l = √(h² + (s/2)²)
Pythagorean theorem using height and half the base edge.
Surface area
SA = s² + 2sl
Base area plus the four triangular side faces.

Your Results

Calculated
Volume
-
V = ⅓ × base area × height
Slant Height
-
l = √(h² + (s/2)²)
Lateral Surface Area
-
Area of the four triangular faces
Total Surface Area
-
Base area + lateral surface area

Ready

Enter a base edge length and height, then press Calculate.

Formula and Method for Square Pyramid Volume

A square pyramid is a solid with a square base and four triangular faces that meet at a single apex directly above the base's center. Its volume is one-third of the base area times the perpendicular height: V = (1/3)s²h, where s is the base edge length and h is the vertical height from the base to the apex. This calculator also derives the slant height and both surface area measurements from the same two inputs.

How the calculation works

Enter the base edge length (s) and the vertical height (h) in the same unit, then choose that unit. The base area is s², and the volume is one-third of the base area times the height: V = (1/3)s²h — the general volume formula for any pyramid or cone, "one-third times base area times height." The slant height (l), the distance from the apex down the middle of a triangular face to the midpoint of a base edge, is found with the Pythagorean theorem: l = √(h² + (s/2)²), because h, half the base edge (s/2), and l form a right triangle. Each of the four triangular faces has area (1/2)·s·l, so the lateral surface area is 2sl, and the total surface area adds the base: SA = s² + 2sl.

Common mistakes

  • Height vs. slant height: h is the straight vertical distance from apex to base center; l (slant height) runs down a face and is always longer. Using the wrong one in the wrong formula gives incorrect volume or surface area.
  • Forgetting the one-third: a square pyramid's volume is one-third of the prism with the same base and height, not the full base area times height (that would be a cuboid).
  • Mixing units: keep the base edge and height in the same unit before entering them — convert inches to feet, or centimeters to meters, first.

Real-world applications

  • Architecture and construction use pyramid volume to estimate material (concrete, fill, roofing) for pyramidal roofs, spires, and monuments.
  • Packaging design uses volume to size pyramidal boxes or containers and estimate their capacity.
  • Engineering and surveying use slant height and surface area to estimate the material needed to clad or panel a sloped, four-sided structure.
  • Geometry and physics courses use square pyramid volume to introduce the general "one-third base times height" rule shared by all cones and pyramids.

Frequently Asked Questions

What is the formula for the volume of a square pyramid?
Volume equals one-third of the base area times the height: V = (1/3) × s² × h, where s is the base edge length and h is the vertical height from the base to the apex. A pyramid with a 6 ft base edge and 8 ft height has a volume of (1/3) × 36 × 8 = 96 ft³.
How do you find the slant height of a square pyramid?
Slant height is l = √(h² + (s/2)²), found with the Pythagorean theorem using the vertical height and half the base edge, since the slant height, the height, and half the base edge form a right triangle.
How do you calculate the surface area of a square pyramid?
Total surface area equals the base area plus the four triangular faces: SA = s² + 2sl, where l is the slant height. The 2sl term is the combined lateral surface area of the four triangles.
What is the difference between height and slant height?
Height (h) is the perpendicular distance from the apex straight down to the center of the base, used in the volume formula. Slant height (l) is the distance from the apex down the middle of a triangular face to the midpoint of a base edge, used in the surface area formula. Slant height is always longer than height for h > 0.

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