Perimeter of Polygons

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SSAT Upper Level Quantitative › Perimeter of Polygons

Questions 1 - 10

The base length of a parallelogram is 10 inches and the side length is 6 inches. Give the perimeter of the parallelogram.

Explanation

Like any polygon, the perimeter of a parallelogram is the total distance around the outside, which can be found by adding together the length of each side. In case of a parallelogram, each pair of opposite sides is the same length, so the perimeter is twice the base plus twice the side length. Or as a formula we can write:

Where:

is the base length of the parallelogram and is the side length. So we can write:

The base length of a parallelogram is 10 inches and the side length is 6 inches. Give the perimeter of the parallelogram.

Explanation

Where:

is the base length of the parallelogram and is the side length. So we can write:

The diagonal of a square is $6\sqrt{2} : cm$ . Find the perimeter of the square.

$24\sqrt{2}:cm^2$

$12\sqrt{2}:cm^2$

Explanation

The diagonal of a square is also the hypotenuse of a triangle whose legs are two sides of the square. Using that information, we can find the length of each side of the square.

$(6\sqrt{2})^2=(\text{side})^2+(\text{side})^2$

$72=2(\text{side})^2$

$36=\text{side}^2$

$6: cm=\text{side}$

Now, multiply the side length by 4 to find the perimeter.

$6:cm\times4:cm=24:cm^2$

Track

The track at Harriet Beecher Stowe High School is a perfect square of with diagonal 600 feet, and is shown in the above figure. Beginning at point A, Corinne runs around the track clockwise three complete times, then contunes to run until she reaches point B. Which of the following comes closest to the distance Corinne runs?

You will need to know that $\sqrt {2} \approx 1.4$ .

$1 \textup{ mi}$

$1\frac{1}{2} \textup{ mi}$

$2\frac{1}{2} \textup{ mi}$

$2 \textup{ mi}$

$\frac{1}{2} \textup{ mi}$

Explanation

A square with diagonal 600 feet will have as its sidelength

$600 \div \sqrt {2 } \approx 600 \div 1.4 = 429$ feet.

If Corinne runs the entire perimeter of the square three times, and then runs on to Point B, she will run this distance a total of about $14 \frac{3}{4}$ times. This is a total of

$429 \times 14 \frac{3}{4} \approx 6,328$ feet.

Divide by 5,280 to convert to miles:

$6,328 \div 5,280 \approx 1.2$

The closest response is 1 mile.

Track

You will need to know that $\sqrt {2} \approx 1.4$ .

$1 \textup{ mi}$

$1\frac{1}{2} \textup{ mi}$

$2\frac{1}{2} \textup{ mi}$

$2 \textup{ mi}$

$\frac{1}{2} \textup{ mi}$

Explanation

A square with diagonal 600 feet will have as its sidelength

$600 \div \sqrt {2 } \approx 600 \div 1.4 = 429$ feet.

If Corinne runs the entire perimeter of the square three times, and then runs on to Point B, she will run this distance a total of about $14 \frac{3}{4}$ times. This is a total of

$429 \times 14 \frac{3}{4} \approx 6,328$ feet.

Divide by 5,280 to convert to miles:

$6,328 \div 5,280 \approx 1.2$

The closest response is 1 mile.

A hexagon with perimeter 60 has four congruent sides of length . Its other two sides are congruent to each other. Give the length of each of those other sides in terms of .

Explanation

The perimeter of a polygon is the sum of the lengths of its sides. Let:

Length of one of those other two sides

Now we can set up an equation and solve it for in terms of :

$\Rightarrow 2x+4t+4=60$

$\Rightarrow 2x=60-4-4t$

$\Rightarrow 2x=56-4t$

$\Rightarrow x=28-2t$

Track

The track at Frederick Douglass High School is a perfect square of with diagonal 400 feet, and is shown in the above figure. Julia wants to run around the track for one mile. If Julia starts at point A and runs clockwise, where will she be after she has run for a mile?

A hint: $\sqrt {2} \approx 1.414$

Explanation

A square with diagonal 400 feet will have as its sidelength

$400 \div \sqrt {2 } \approx 400 \div 1.414 = 283$ feet.

Julia wants to run one mile, or 5,280 feet; this will be

$5,280 \div 283 \approx 18.7$ sidelengths.

Julia will run around the track four times, then another 2 sidelengths. She will then run seven-tenths of the length of the "bottom" side, ending up at point D.

Track

A hint: $\sqrt {2} \approx 1.414$

Explanation

A square with diagonal 400 feet will have as its sidelength

$400 \div \sqrt {2 } \approx 400 \div 1.414 = 283$ feet.

Julia wants to run one mile, or 5,280 feet; this will be

$5,280 \div 283 \approx 18.7$ sidelengths.

Julia will run around the track four times, then another 2 sidelengths. She will then run seven-tenths of the length of the "bottom" side, ending up at point D.

A hexagon with perimeter 60 has four congruent sides of length . Its other two sides are congruent to each other. Give the length of each of those other sides in terms of .

Explanation

The perimeter of a polygon is the sum of the lengths of its sides. Let:

Length of one of those other two sides

Now we can set up an equation and solve it for in terms of :

$\Rightarrow 2x+4t+4=60$

$\Rightarrow 2x=60-4-4t$

$\Rightarrow 2x=56-4t$

$\Rightarrow x=28-2t$

The diagonal of a square is $6\sqrt{2} : cm$ . Find the perimeter of the square.

$24\sqrt{2}:cm^2$

$12\sqrt{2}:cm^2$

Explanation

The diagonal of a square is also the hypotenuse of a triangle whose legs are two sides of the square. Using that information, we can find the length of each side of the square.

$(6\sqrt{2})^2=(\text{side})^2+(\text{side})^2$

$72=2(\text{side})^2$

$36=\text{side}^2$

$6: cm=\text{side}$

Now, multiply the side length by 4 to find the perimeter.

$6:cm\times4:cm=24:cm^2$

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