How cone volume and surface area formulas work
Where the one third volume rule, the slant height, and the lateral area formula come from, with a worked radius 3 height 4 example.
The one third volume rule
Picture a cylinder and a cone standing on the same circular base with the same height. If you filled the cone with water and poured it into the cylinder, it would take exactly three cones to fill the cylinder. That is the meaning of the one third in the volume formula, V equals one third times pi times r squared times h. The pi times r squared part is the area of the base, and multiplying by the height then dividing by three gives the space the cone encloses.
Slant height and the Pythagorean theorem
The slant height is not the same as the vertical height. It runs along the outside of the cone from the base rim to the tip. Because the radius, the vertical height and the slant form a right triangle, the slant equals the square root of the radius squared plus the height squared. A radius of 3 with a height of 4 produces a slant of exactly 5, the familiar 3-4-5 triangle, which is why that example is so often used in class.
Unrolling the curved side
If you cut the curved wall of a cone and lay it flat, it forms a sector of a larger circle whose radius is the slant height. The area of that sector reduces neatly to pi times the base radius times the slant height, which is the lateral area. Adding the flat circular base, pi times the radius squared, gives the total surface area. Factoring that sum produces the compact form pi times r times the quantity r plus slant.
A full worked example
Take radius 3 and height 4. The slant is the square root of 9 plus 16, which is 5. The volume is one third of pi times 9 times 4, giving 37.699112 cubic units. The lateral area is pi times 3 times 5, which rounds to 47.12389 square units. Adding the base area of 9 pi produces a total surface area of 75.398224 square units, all of which the calculator returns the moment you type the two inputs.