Frustum formulas explained step by step
Where the truncated cone volume, slant height and surface area formulas come from, and how to apply them by hand with a worked example.
Why a frustum needs its own volume formula
You cannot find a frustum's volume by averaging the two end circles, because the radius changes smoothly along the height and the cross section grows with the square of the radius. The clean way is to picture the full cone the frustum came from, then subtract the small cone that was sliced off the top. When you carry out that subtraction and simplify, the two cone volumes combine into a single expression that depends only on the two radii and the height. That expression is (1/3) pi h (rb squared + rb rt + rt squared), which is exactly what this calculator evaluates. The middle term, rb times rt, is what stops the answer from being a simple average and captures the taper.
Slant height versus vertical height
The height in the volume formula is the perpendicular distance between the two flat ends, measured straight up the central axis. The slant height is different: it is the diagonal distance along the sloped outer surface from a point on the bottom rim to the point directly above it on the top rim. Because the radii differ, that slope leans inward by an amount equal to rb minus rt. A right triangle with legs h and (rb - rt) gives the slant as sqrt(h squared + (rb - rt) squared). For the bottom 4, top 2, height 6 frustum that is sqrt(36 + 4), which rounds to 6.324555. Mixing these two heights up is the most common frustum mistake.
Building up the surface area
Surface area comes in two parts. The curved side unrolls into a flat ring shape whose area works out to pi times the sum of the radii times the slant height, giving the lateral area. For the same worked example that is pi times 6 times 6.324555, or about 119.215059. If your object is closed, you then add the two circular ends, pi times rb squared and pi times rt squared, to reach the total surface area of about 182.046912. Choosing between lateral and total area is a modeling decision: an open lampshade uses lateral area, while a sealed tank uses total.
A worked example from start to finish
Take a planter with a bottom radius of 5, a top radius of 3 and a height of 10. The volume is (1/3) pi times 10 times (25 + 15 + 9), which is (1/3) pi times 10 times 49, giving about 513.1268 cubic units. The slant height is sqrt(100 + 4), or about 10.198039. The lateral area is pi times 8 times that slant, roughly 256.304676, and adding the two end circles of 25 pi and 9 pi brings the total surface area to about 363.118826. Those four numbers match the calculator exactly, so you can use the hand method to sanity check any result.