Formulas for Surface Area, Volume, and Height of Common 3D Shapes

Surface Area and Volume Formulas

Ever wonder how to calculate the amount of wrapping paper needed for a box or how much water a container can hold? That's where surface area and volume come in!

For a rectangular prism, the surface area equation is 2(lw) + 2(lh) + 2(wh), where l is length, w is width, and h is height. Its volume is simply V = lwh. If you know the volume and base dimensions, you can find the height using h = V/(lw).

A cylinder has surface area SA = 2πr² + 2πr$h$, which includes the two circular ends plus the curved side. Its volume is V = πr²h, where r is radius and h is height.

Pro Tip: Remember that the radius $r$ is always half the diameter. When a problem gives you the diameter, divide by 2 before plugging into your formulas!

For a cube (where all sides have equal length s), things simplify nicely: surface area is SA = 6s² and volume is V = s³. A triangular prism has surface area SA = 2b$$h + (P)$h$ and volume V = b$$h, where P is the perimeter of the base.

For shapes that come to a point like pyramids, the surface area is SA = b² + ½(P)(Sh), where Sh is the slant height. Their volume is V = ⅓b$$h, and you can find height with h = 3V/base area. Similarly, a cone has surface area SA = πr² + πr(Sh) and volume V = ⅓πr²h.