Pyramid Calculator
Volume24.000 ft³
Surface Area62.604 ft²
Slant Height3.606 ft
This calculator works out the volume, surface area, and slant height of a rectangular-base pyramid from its base length, base width, and vertical height. A pyramid tapers from a flat rectangular base up to a single apex, so its volume is exactly one third of the box that would enclose it.
Formula
V = (1/3)·L·W·h; slant = √(h² + (W/2)²); SA = L·W + L·slantW + W·slantL
- L
- Base length
- W
- Base width
- h
- Vertical (perpendicular) height from base centre to apex
How it works
- Enter the base length, the base width, and the vertical height measured straight up from the centre of the base to the apex.
- Volume is one third of base length × base width × height. The total surface area adds the rectangular base to the four triangular faces, using a slant height for each pair of opposite faces. Results are rounded to three decimals.
Worked example
A pyramid has a square base 6 by 6 and a height of 4.
- Volume = (1/3) × 6 × 6 × 4 = (1/3) × 144 = 48.
- Slant height = √(4² + (6/2)²) = √(16 + 9) = √25 = 5.
- Surface area = 6×6 + 6×5 + 6×5 = 36 + 30 + 30 = 96.
Volume = 48 cubic units, surface area = 96 square units, slant height = 5.
Frequently asked questions
- What shape of pyramid does this calculator handle?
- It handles a right pyramid with a rectangular base, where the apex sits directly above the centre of the base. Square bases are simply the case where length and width are equal.
- What is the difference between vertical height and slant height?
- Vertical height is the straight-up distance from the base centre to the apex, the value you enter. Slant height runs along a triangular face from the base edge to the apex and is longer, found from the height and half the base side.
- Why is the volume one third of length times width times height?
- A pyramid fills exactly one third of the rectangular prism that shares its base and height. That is why the volume formula multiplies the base area by the height and then by one third.