Pyramid Calculator
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The volume of any pyramid is one third of the base area times the height: V = ⅓Bh. A square pyramid with a 6 × 6 base and height 4 has volume ⅓ × 36 × 4 = 48, slant height 5 and total surface area 96.
Sketch not to scale for the depth direction.
| Volume and area | |
| Volume | 48 |
| Total surface area | 96 |
| Lateral surface area | 60 |
| Base area | 36 |
| Lengths | |
| Height h | 4 |
| Slant height ℓ | 5 |
| Lateral edge (base corner to apex) | 5.830951895 |
| Base perimeter | 24 |
Show steps
- Base: square, area B = 6² = 36.
- Volume: \(V = \frac{1}{3}Bh\) = ⅓ × 36 × 4 = 48.
- Slant height: \(\ell = \sqrt{h^{2} + r^{2}}\) where r = 3 is the base apothem: ℓ = 5.
- Lateral area = ½ × perimeter × ℓ = ½ × 24 × 5 = 60.
- Total surface area = base + lateral = 36 + 60 = 96.
How to use the pyramid calculator
- Pick the base shape and enter its side length (and width for a rectangle).
- Enter the height, or switch to slant height if that is what you know.
- Press Calculate for the volume, total and lateral surface area, slant height and edge length.
Pyramid formulas
| Quantity | Formula |
|---|---|
| Volume | V = ⅓ × base area × height |
| Slant height (regular base) | ℓ = √(h² + r²), r = base apothem |
| Lateral area (regular base) | ½ × base perimeter × ℓ |
| Total surface area | base area + lateral area |
| Lateral edge | √(h² + R²), R = center-to-corner distance |
For a square base of side s, r = s/2, so ℓ = √(h² + s²/4).
Worked example
Square base 6, height 4. Base area 36, so V = ⅓ × 36 × 4 = 48. Slant height \(\ell = \sqrt{4^{2} + 3^{2}} = 5\). Lateral area = ½ × 24 × 5 = 60. Total surface area = 36 + 60 = 96.
Why one third?
A cube can be cut into three identical square pyramids that meet at one corner. Each has the cube’s base and height, so each holds exactly one third of the cube’s volume. The same ⅓ appears for cones.
Pyramid practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do you find the volume of a pyramid?
Find the area of the base, multiply by the height, then divide by 3.
What is the difference between height and slant height?
Height is straight down from the apex to the base. Slant height runs down the middle of a triangular face. Slant height is always longer.
How do you find the surface area of a square pyramid?
Base area s² plus four triangles of area ½ × s × ℓ: SA = s² + 2sℓ.
Does the formula work for an oblique pyramid?
The volume formula ⅓Bh works for any pyramid if h is the perpendicular height. Surface area formulas here assume the apex is over the center of the base.
What units does the answer use?
The same unit you enter: lengths in that unit, areas in square units and volume in cubic units.
Sources
- OpenStax Prealgebra 2e: Solve Geometry Applications: Volume and Surface Area
- Wolfram MathWorld: Pyramid
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