Square pyramid volume and surface area explained
How the volume, slant height, lateral area and total surface area of a right square pyramid are derived, with worked numbers you can check.
The one third rule for volume
Every pyramid holds one third of the volume of a prism built on the same base with the same height. For a square base that means volume is base edge squared times height divided by three. A pyramid with a 6 by 6 base and a height of 4 has a base area of 36, so its volume is 36 times 4 over 3, which is 48 cubic units. The same rule powers cone volume, where the square base is simply swapped for a circle.
Finding the slant height
The slant height is the distance from the apex down the centre of a triangular face, and it is what you need for surface area rather than the vertical height. Picture a right triangle whose legs are the vertical height and half the base edge. The slant height is the hypotenuse, so it equals the square root of the height squared plus half the base edge squared. With a height of 4 and a base edge of 6 the legs are 4 and 3, giving a clean slant height of 5.
Lateral area versus total surface area
A square pyramid has four identical triangular faces. Each triangle has a base equal to the base edge and a height equal to the slant height, so all four together come to two times the base edge times the slant height. That is the lateral area, the part you would paint or wrap. Adding the square base underneath gives the total surface area, which for a base edge of 6 and slant height of 5 is 36 plus 60, or 96 square units.
Checking a worked example
Take a base edge of 10 and a height of 12. The base area is 100 and the volume is 100 times 12 over 3, which is 400. The slant height is the square root of 12 squared plus 5 squared, and since 5, 12 and 13 form a right triangle the slant height is exactly 13. The lateral area is two times 10 times 13, or 260, and adding the 100 base gives a total surface area of 360. These are the same figures the calculator returns.