SSAT Upper Level Quantitative - How to find the perimeter of a rectangle

A rectangle has length 30 inches and width 25 inches. Which of the following is true about its perimeter?

Its perimeter is between 9 and 10 feet.

Its perimeter is between 10 and 11 feet.

Its perimeter is between 8 and 9 feet.

Its perimeter is between 4 and 5 feet.

Its perimeter is between 5 and 6 feet.

Explanation

In inches, the perimeter of the rectangle can be calculated by substituting in the following formula:

$P = 2 (30 + 25) = 2\times 55 = 110$

The perimeter is 110 inches.

Now divide by 12 to convert to feet:

$110 \div 12 = 9 \textrm{ R }2$

This makes the perimeter 9 feet 2 inches, which is between 9 feet and 10 feet.

Which of these polygons has the same perimeter as a rectangle with length 55 inches and width 15 inches?

The other answer choices are incorrect.

A regular pentagon with sidelength two feet

A regular heptagon with sidelength two feet

A regular hexagon with sidelength two feet

A regular octagon with sidelength two feet

Explanation

The perimeter of a rectangle is twice the sum of its length and its width; a rectangle with dimensions 55 inches and 15 inches has perimeter

inches.

All of the polygons in the choices are regular - that is, all have congruent sides - and all have sidelength two feet, or 24 inches, so we divide 140 by 24 to determine how many sides such a polygon would need to have a perimeter equal to the rectangle. However,

$140 \div 24 = 5 \frac{5}{6}$ ,

so there cannot be a regular polygon with these characteristics. All of the choices fail, so the correct response is that none are correct.

The length and width of a rectangle are and , respectively. Give its perimeter in terms of .

Explanation

The perimeter of a rectangle is , where is the length and is the width of the rectangle. In order to find the perimeter we can substitute the and in the perimeter formula:

The length of a rectangle is and the width of this rectangle is meters shorter than its length. Give its perimeter in terms of .

Explanation

The length of the rectangle is known, so we can find the width in terms of :

The perimeter of a rectangle is , where is the length and is the width of the rectangle.

In order to find the perimeter we can substitute the and in the perimeter formula:

A rectangle has a length of inches and a width of inches. Which of the following is true about the rectangle perimeter if ?

Its perimeter is between 7 and 8 feet.

Its perimeter is between 8 and 9 feet.

Its perimeter is less than 7 feet.

Its perimeter is more than 8 feet.

Its perimeter is between 7.2 and 7.4 feet.

Explanation

Substitute to get and :

$W=5t=5\times 4=20$

The perimeter of a rectangle is , where is the length and is the width of the rectangle. So we have:

$P=2L+2W=2\times 25+2\times20=50+40=90$ inches

Now we should divide the perimeter by 12 in order to convert to feet:

$Perimeter=90\div 12=7.5$ feet

So the perimeter is 7 feet and 6 inches, which is between 7 and 8 feet.

The length and width of a rectangle are and . Give its perimeter in terms of .

$2x^{2} -25x -112$

$2x^{2} +25x -112$

Explanation

A rectangle has perimeter , the length and the width. Substitute and in the perimeter formula, and simplify.

$= 2 \left ( x+16\right ) + 2 \left ( 2x-7\right )$

$= 2 \cdot x+2 \cdot 16 +2 \cdot 2x-2 \cdot 7$

Perimeter of a rectangle is 36 inches. If the width of the rectangle is 3 inches less than its length, give the length and width of the rectangle.

inches

Explanation

Let:

$Length=x\Rightarrow Width=x-3$

The perimeter of a rectangle is , where is the length and is the width of the rectangle. The perimeter is known so we can set up an equation in terms of and solve it: