How to find the surface area of a prism

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ISEE Upper Level Quantitative Reasoning › How to find the surface area of a prism

Questions 1 - 3

1

Find the surface area of a non-cubic prism with the following measurements:

Explanation

The surface area of a non-cubic prism can be determined using the equation:

2

Box

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

$4x^{2} + 30 x$

$15x^{2}$

Explanation

A square has four sides of equal length, as seen in the diagram below.

Box

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces): $15 \cdot 15 =225$

The total surface area: $4 \cdot 15x + 2 \cdot 225 = 60x + 450$

3

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

(A) is greater

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

Explanation

The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

$S = 2( 16t \cdot 9t +9t \cdot t +16t \cdot t)$

$S = 2( 144t ^{2}+9t^{2} +16t^{2})$

$S = 2( 169t ^{2})$

$S = 338t ^{2}$

$S = 2( 8t \cdot 6t +6t \cdot 3t +8t \cdot 3 t)$

$S = 2( 48t^{2} +18t ^{2}+24 t^{2})$

$S = 2( 90 t^{2})$

$S = 180 t^{2}$

Regardless of the value of , $338t ^{2} > 180t ^{2}$ - that is, the first prism has the greater surface area. (A) is greater.

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