How cone volume and surface area work
A cone's volume comes directly from its base radius and height. Surface area first needs the slant height — the diagonal distance from the apex to the base's edge — then adds the base and the unrolled side.
- r — base radius
- h — cone height
- l — slant height (hypotenuse)
- V — volume
- S — total surface area (base + side)
More detail
Why slant height matters
Unroll a cone's curved side and it becomes a flat sector whose radius is the slant height (l) — not the height (h). Since l is the actual hypotenuse from the apex to the base's rim, it must be found first via the Pythagorean relation l=√(r²+h²) before the side area (πrl) can be computed.
Quick check. A cone with radius 3 and height 4 has a slant height of exactly 5 (the 3-4-5 right triangle) — volume works out to about 37.70 and surface area to about 75.40, a fast way to sanity-check the calculator.
Frequently asked questions
What is the volume of a cone with radius 3 and height 4?
Plugging into V=(1/3)πr²h gives V=(1/3)×π×3²×4=12π≈37.70. The slant height is l=√(3²+4²)=5, the classic 3-4-5 right triangle, which keeps the numbers clean.
What's the difference between slant height (l) and height (h)?
Height (h) is the perpendicular distance from the apex to the center of the base; slant height (l) is the hypotenuse from the apex to a point on the base's rim, always longer than h and found via l=√(r²+h²). The surface area formula (S=πr²+πrl) always uses l, never h.
Why does the result show "—" when radius or height is 0?
If either value is 0 or negative, the shape either has zero volume or isn't a real cone at all, so the calculator only returns a result when both radius and height are greater than 0.
Why is there a 1/3 in the cone volume formula?
A cone's volume is exactly one-third the volume of a cylinder (πr²h) with the same base and height — a standard solid-geometry result provable by integration or Cavalieri's principle.
How do the base and side add up to the total surface area?
In S=πr²+πrl, πr² is the area of the circular base and πrl is the area of the unrolled sector forming the side. Adding them gives πr(r+l), the cone's total surface area.