How do you find the volume and surface area of a rectangular prism (box)?

Category: 3D Shapes »

Course: Geometry »

 
Relevant MM Lessons:
  • Forthcoming

Boxes

As long as the box you're working with truly fits the definition of a box, meaning it is built of rectangles on all sides and bases and is not slanted, then it has three measures: a length, a width, and a height.As a three-dimensional object, a rectangular prism's volume is the product of each of its dimensions.$$V = lwh$$Now consider the surface area.A rectangular box has $6$ sides but every side has an equal partner sitting opposite to it. The bottom and top sides each have area $lw$. The lateral sides are each height $h$, with two of them having area $hl$ and the other two having area $hw$.Putting these ideas together, the formula for the surface area of a cube is$$S = 2lw + 2hl + 2hw$$

Tips

  • If you are learning the volume formulas for several shapes, it is helpful to know that volume is always (base area) $\times$ (height) regardless of whether the base is a rectangle
  • A shortcut for remembering the surface area formula is that it is simply the sum of every combination of $h$, $l$, and $w$, with a $2$ in front