Prism basics

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A rectangular prism is a box, and it has six rectangular faces in three matching pairs.
Its volume fills the inside, and its surface area covers the outside.
rectangular_prism.png
The figure shows a prism with its length, width, and height labeled.
The volume is V=length×width×heightV = \text{length} \times \text{width} \times \text{height}.
The surface area is 2(lw+lh+wh)2(lw + lh + wh), since opposite faces have equal area.
For a square base of side xx and height 2x2x, group the faces.
There are 22 square bases, each x2x^2, and 44 rectangular sides, each x×2x=2x2x \times 2x = 2x^2.
So the surface area is 2x2+4(2x2)=10x22x^2 + 4(2x^2) = 10x^2.
You can set surface area or volume equal to a given value to solve for xx.
If 10x2=16010x^2 = 160, then x2=16x^2 = 16, so x=4x = 4.
The volume is then 2x3=2(4)3=1282x^3 = 2(4)^3 = 128.

Worked examples

A box has length 33, width 44, and height 55. What is its volume?
Volume is length times width times height: 3×4×53 \times 4 \times 5.
So the volume is 6060.
A prism has a square base of side xx and height 2x2x. What is its surface area?
Two square bases give 2x22x^2, and four side faces give 4×(x×2x)=8x24 \times (x \times 2x) = 8x^2.
So the surface area is 2x2+8x2=10x22x^2 + 8x^2 = 10x^2.
A prism with square base side xx and height 2x2x has surface area 9090. Find its volume.
Set 10x2=9010x^2 = 90, so x2=9x^2 = 9 and x=3x = 3.
The volume is 2x3=2(3)3=542x^3 = 2(3)^3 = 54.

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