The four formulas
A rectangular prism, a box, has volume length × width × height. Its surface area is the sum of the areas of all six rectangular faces: 2(lw + lh + wh), since opposite faces come in matching pairs.
A cylinder's volume is πr²h: the area of the circular base, πr², times the height. Its total surface area covers both circular ends plus the curved side wrapped around them, 2πr² + 2πrh, usually written 2πr(r + h).
A sphere's volume is (4/3)πr³, and its surface area is 4πr², a result attributed to Archimedes: a sphere's surface area equals exactly four times the area of its largest circular cross-section.
A cone's volume is (1/3)πr²h, one third of a cylinder sharing the same base and height. Its total surface area is the flat base circle plus the slanted side: πr² + πrl, where the slant height l is √(r² + h²) by the Pythagorean theorem, since the slant is the hypotenuse of a right triangle formed by the radius and the height.
Worked examples
A shipping box 2 ft long, 1.5 ft wide, and 1 ft tall: volume = 2 × 1.5 × 1 = 3 cubic feet. Surface area = 2(2×1.5 + 2×1 + 1.5×1) = 2(3 + 2 + 1.5) = 13 square feet, the amount of cardboard the box is made from.
A cylindrical drum with a 2 ft radius and 6 ft height: volume = π × 2² × 6 = 24π ≈ 75.4 cubic feet. Surface area = 2π × 2 × (2 + 6) = 32π ≈ 100.5 square feet, counting the top and bottom lids plus the wrapped side.
Radius, not diameter
Every formula above uses the radius, the distance from the center to the edge. Product listings and everyday measurements more often give the diameter, the distance all the way across, so if that is what is on hand, halve it before entering a value here.
The cone result is the total surface area: base plus the slanted side. A party hat or a paper cone rolled with no bottom only has the slanted portion, so subtract πr² from the total shown here to get that lateral-only figure.