SSAT Upper Level Math : Areas and Perimeters of Polygons

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

Example Question #1 : How To Find The Perimeter Of A Hexagon

A regular hexagon has perimeter 9 meters. Give the length of one side in millimeters.

Possible Answers:

\(\displaystyle 1,200\textrm{ mm }\)

\(\displaystyle 1,500 \textrm{ mm }\)

\(\displaystyle 2,400 \textrm{ mm }\)

\(\displaystyle 2,000\textrm{ mm }\)

\(\displaystyle 1,800\textrm{ mm }\)

Correct answer:

\(\displaystyle 1,500 \textrm{ mm }\)

Explanation:

One meter is equal to 1,000 millimeters, so the perimeter of 9 meters can be expressed as:

9 meters = \(\displaystyle 9 \times 1,000 = 9,000\) millimeters.

Since the six sides of a regular hexagon are congruent, divide by 6:

\(\displaystyle 9,000 \div 6 = 1,500\) millimeters.

Example Question #1 : Areas And Perimeters Of Polygons

A hexagon with perimeter 60 has four congruent sides of length \(\displaystyle t+1\). Its other two sides are congruent to each other. Give the length of each of those other sides in terms of \(\displaystyle t\).

Possible Answers:

\(\displaystyle 56+2t\)

\(\displaystyle 28-t\)

\(\displaystyle 28-2t\)

\(\displaystyle 56-2t\)

\(\displaystyle 28+2t\)

Correct answer:

\(\displaystyle 28-2t\)

Explanation:

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

\(\displaystyle x=\) Length of one of those other two sides

Now we can set up an equation and solve it for \(\displaystyle x\) in terms of \(\displaystyle t\):

\(\displaystyle 2x+4(t+1)=60\)

\(\displaystyle \Rightarrow 2x+4t+4=60\)

\(\displaystyle \Rightarrow 2x=60-4-4t\)

\(\displaystyle \Rightarrow 2x=56-4t\)

\(\displaystyle \Rightarrow x=28-2t\)

 

Example Question #2 : Areas And Perimeters Of Polygons

Two sides of a hexagon have a length of \(\displaystyle t\), two other sides have the length of \(\displaystyle t-1\), and the rest of the sides have the length of \(\displaystyle t+1\). Give the perimeter of the hexagon.

Possible Answers:

\(\displaystyle 6t+6\)

\(\displaystyle 8t+8\)

\(\displaystyle 8t\)

\(\displaystyle 6t-6\)

\(\displaystyle 6t\)

Correct answer:

\(\displaystyle 6t\)

Explanation:

The perimeter of a polygon is the sum of the lengths of its sides. So we can write:

\(\displaystyle Perimeter=2\left [ t+(t+1)+(t-1) \right ]=2(3t)=6t\)

Example Question #2 : How To Find The Perimeter Of A Hexagon

A regular hexagon has perimeter 15 feet. Give the length of one side in inches.

Possible Answers:

\(\displaystyle 30\)

\(\displaystyle 18\)

\(\displaystyle 24\)

\(\displaystyle 36\)

\(\displaystyle 15\)

Correct answer:

\(\displaystyle 30\)

Explanation:

As the six sides of a regular hexagon are congruent, we can write:

\(\displaystyle Perimeter=6a=15\Rightarrow a=2.5\) feet; \(\displaystyle a\) is the length of each side.

One feet is equal to 12 inches, so we can write:

\(\displaystyle a=2.5\times 12=30\) inches

Example Question #4 : Perimeter Of Polygons

Each interior angle of a hexagon is 120 degrees and the perimeter of the hexagon is 120 inches. Find the length of each side of the hexagon.

Possible Answers:

\(\displaystyle 25\ inches\)

\(\displaystyle 20\ inches\)

\(\displaystyle 10\ inches\)

\(\displaystyle 30\ inches\)

\(\displaystyle 60\ inches\)

Correct answer:

\(\displaystyle 20\ inches\)

Explanation:

Since each interior angle of a hexagon is 120 degrees, we have a regular hexagon with identical side lengths. And we know that the perimeter of a polygon is the sum of the lengths of its sides. So we can write:

\(\displaystyle Perimeter=6a=120\Rightarrow a=20\) inches

Example Question #5 : Perimeter Of Polygons

A hexagon with perimeter of 48 has three congruent sides of \(\displaystyle 2t+3\). Its other three sides are congruent to each other with the length of \(\displaystyle 2t-7\). Find \(\displaystyle t\).

Possible Answers:

\(\displaystyle 6\)

\(\displaystyle 5\)

\(\displaystyle 3\)

\(\displaystyle 4\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle 5\)

Explanation:

The perimeter of a polygon is the sum of the lengths of its sides. Since three sides are congruent with the length of \(\displaystyle 2t+3\) and the rest of the sides have the length of \(\displaystyle 2t-7\) we can write:

 

\(\displaystyle Perimeter=3(2t+3)+3(2t-7)=48\)

Now we should solve the equation for \(\displaystyle t\):

\(\displaystyle 6t+9+6t-21=48\Rightarrow 12t-12=48\Rightarrow 12t=60\Rightarrow t=5\)

Example Question #742 : Geometry

A regular pentagon has sidelength one foot; a regular hexagon has sidelength ten inches. The perimeter of a regular octagon is the sum of the perimeters of the pentagon and the hexagon. What is the measure of one side of the octagon?

Possible Answers:

\(\displaystyle 16 \textrm{ in }\)

\(\displaystyle 10 \textrm{ in }\)

\(\displaystyle 15 \textrm{ in }\)

\(\displaystyle 12 \textrm{ in }\)

\(\displaystyle 18 \textrm{ in }\)

Correct answer:

\(\displaystyle 15 \textrm{ in }\)

Explanation:

A regular polygon has all of its sides the same length. The pentagon has perimeter \(\displaystyle 5 \times 12 \textrm{ in } = 60 \textrm{ in }\); the hexagon has perimeter \(\displaystyle 6 \times 10 \textrm{ in } = 60 \textrm{ in }\). The sum of the perimeters is \(\displaystyle 60 \textrm{ in } + 60 \textrm{ in }= 120 \textrm{ in }\), which is the perimeter of the octagon; each side of the octagon has length \(\displaystyle 120 \textrm{ in } \div 8 = 15 \textrm{ in }\).

Example Question #1 : Areas And Perimeters Of Polygons

Find the perimeter of a hexagon with a side length of \(\displaystyle 6a+b\).

Possible Answers:

\(\displaystyle 6a+6b\)

\(\displaystyle 12a+6b\)

\(\displaystyle 36a+6b\)

\(\displaystyle 36a+36b\)

\(\displaystyle 36a+b\)

Correct answer:

\(\displaystyle 36a+6b\)

Explanation:

A hexagon has six sides.  The perimeter of a hexagon is:

\(\displaystyle P=6s\)

Substitute the side length.

\(\displaystyle P=6s=6(6a+b)=36a+6b\)

Example Question #1 : How To Find The Area Of A Hexagon

Find the area of a regular hexagon that has side lengths of \(\displaystyle 6\).

Possible Answers:

\(\displaystyle 48\sqrt3\)

\(\displaystyle 36\sqrt3\)

\(\displaystyle 54\sqrt3\)

\(\displaystyle 36\)

Correct answer:

\(\displaystyle 54\sqrt3\)

Explanation:

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2\).

Now, substitute in the length of the side into this equation.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times6^2\)

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times 36\)

\(\displaystyle \text{Area}=\frac{108\sqrt3}{2}=54\sqrt3\)

 

Example Question #2 : How To Find The Area Of A Hexagon

Find the area of a regular hexagon that has a side length of \(\displaystyle 8\).

Possible Answers:

\(\displaystyle 96\sqrt3\)

\(\displaystyle 192\)

\(\displaystyle 192\sqrt3\)

\(\displaystyle 48\sqrt3\)

Correct answer:

\(\displaystyle 96\sqrt3\)

Explanation:

Use the following formula to find the area of a regular hexagon:

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2\).

Now, substitute in the length of the side into this equation.

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times 8^2\)

\(\displaystyle \text{Area}=\frac{3\sqrt3}{2}\times 64\)

\(\displaystyle \text{Area}=\frac{192\sqrt3}{2}=96\sqrt3\)

 

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