Volume, Surface Area, and Diagonal of a Rectangular Box

When packing items in delivery boxes, or choosing the size of cabinets and dressers, it is helpful to know the volume, surface area, and diagonal length of a rectangular prism, aka rectangular box.

The volume of a box is a measure of its capacity. Surface area is helpful to know in cases when you need to cover a box, such as wrapping a gift. For rectangular bookcases and cabinets, the length of the diagonal tells you the minimum vertical clearance needed to safely move furniture in a room with a low ceiling.

The formulas below show how to compute the volume V, surface area S, and diagonal D if you know the length L, width W, and height H. You can also use the calculator on the left to compute the volume, surface area, and diagonal.

The box calculator will also tell you the dimensions of the box if you input the values for V, S, and D. For example, if a box has a volume of 4000 cubic inches, a surface area of 1600 square inches, and a diagonal of 30 inches, then its dimensions are 20, 20, and 10.

Formulas

V = WLH

S = 2(WL + WH + LH)

D = sqrt(W² + L² + H²)

Example: A shipping box is 24 inches long, 18 inches wide, and 20 inches tall. Its volume is then (18)(24)(20) = 8640 cubic inches, or 5 cubic feet. Its surface area is 2(432+360+480) = 2544 square inches, or 17.67 square feet. The length of the diagonal is sqrt(324+576+400) = 36.05 inches, or 3 feet.

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