Question

What will happen to the surface area of a rectangular prism if all three of its dimensions are doubled? Tripled? Explain your answer?

Answer 1

`A_("Double")=4A`

`A_("Triple")=9A`

Explanation:

.

The surface area of a prism is equal to the surface areas of the sides plus the surface areas of the base and the top. In a rectangular prism, all six surfaces are rectangles as shown below:

Since the area of a rectangle is equal to length X width, (`l*w`, the surface area of the sides can be calculated by multiplying the perimeter of the base by the height:

Surface Area of sides`=2(l+w)h`

Surface area of the base`=lw`

Surface area of the top`=lw`

Total surface Area =`A=2(l+w)h+2lw=2(lh+wh+lw)`

If we double all three dimensions we get:

`A_("Double")=2[(2l)(2h)+(2w)(2h)+(2l)(2w)]`

`A_("Double")=2(4lh+4wh+4lw)`

`A_("Double")=4[2(lh+wh+lw)]`

`A_("Double")=4A`

Doubling the dimensions makes the surface area `4` times the original surface area.

`A_("Triple")=2[(3l)(3h)+(3w)(3h)+(3l)(3w)]`

`A_("Triple")=2(9lh+9wh+9lw)`

`A_("Triple")=9[2(lh+wh+lw)]`

`A_("Triple")=9A`

Tripling the dimensions makes the surface area `9` times the original surface area.