Areas and Perimeters of Polygons - SSAT Upper Level Quantitative

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Question

A rectangle with a width of 6 inches has an area of 48 square inches. Give the sum of the lengths of the rectangle's diagonals.

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Answer

A rectangle has two congruent diagonals. A diagonal of a rectangle divides it into two identical right triangles. The diagonal of the rectangle is the hypotenuse of these triangles. We can use the Pythagorean Theorem to find the length of the diagonal if we know the width and height of the rectangle.

$Diagonal=\sqrt{w^2+h^2}$

where:

is the width of the rectangle
is the height of the rectangle

First, we find the height of the rectangle:

$h=\frac{area}{w}=\frac{48}{6}=8$

So we can write:

$Diagonal=\sqrt{w^2+h^2}=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$ inches

As a rectangle has two diagonals with the same length, the sum of the diagonals is inches.

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Question

Find the area of a regular hexagon that has side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times6^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 36$

$\text{Area}=\frac{108\sqrt3}{2}=54\sqrt3$

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Question

Find the area of a regular hexagon that has a side length of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times 8^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 64$

$\text{Area}=\frac{192\sqrt3}{2}=96\sqrt3$

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Question

Find the area of a regular hexagon that has a side length of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times 12^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 144$

$\text{Area}=\frac{432\sqrt3}{2}=216\sqrt3$

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Question

Find the area of a regular hexagon that has side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times 7^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 49$

$\text{Area}=\frac{147}{2}\sqrt3$

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Question

Find the area of a regular hexagon that has side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times 10^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 100$

$\text{Area}=\frac{300\sqrt3}{2}=150\sqrt3$

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Question

Find the area of a regular hexagon with side lengths .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times 20^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 400$

$\text{Area}=\frac{1200\sqrt3}{2}=600\sqrt3$

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Question

Find the area of a regular hexagon that has side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times x^2$

$\text{Area}=\frac{3x^2}{2}\sqrt3$

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Question

Find the area of a regular hexagon with side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times (2t)^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 4t^2$

$\text{Area}=\frac{12t^2\sqrt3}{2}=6t^2\sqrt3$

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Question

Find the area of a regular hexagon with side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times (3m)^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 9m^2$

$\text{Area}=\frac{27m^2}{2}\sqrt3$

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Question

Find the area of a regular hexagon with side lengths of $\frac{1}{2}$ .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times \left(\frac{1}{2}\right)^2$

$\text{Area}=\frac{3\sqrt3}{2}\times \frac{1}{4}$

$\text{Area}=\frac{3\sqrt3}{8}$

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Question

Find the area of a regular hexagon with side lengths of .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

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

$\text{Area}=\frac{3\sqrt3}{2}\times 64n^2$

$\text{Area}=\frac{192n^2\sqrt3}{2}=96n^2\sqrt3$

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Question

Find the area of a regular hexagon with side lengths .

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Answer

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

$\text{Area}=\frac{3\sqrt3}{2}\times \text{side}^2$ .

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

$\text{Area}=\frac{3\sqrt3}{2}\times (12c)^2$

$\text{Area}=\frac{3\sqrt3}{2}\times 144c^2$

$\text{Area}=\frac{432c^2\sqrt3}{2}=216c^2\sqrt3$

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Question

A parallelogram has the base length of and the altitude of . Give the area of the parallelogram.

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Answer

The area of a parallelogram is given by:

Where is the base length and is the corresponding altitude. So we can write:

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Question

A parallelogram has a base length of which is 3 times longer than its corresponding altitude. The area of the parallelogram is 12 square inches. Give the .

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Answer

Base length is so the corresponding altitude is $\frac{t}{3}$ .

The area of a parallelogram is given by:

Where:

is the length of any base
is the corresponding altitude

So we can write:

$12=t\times \frac{t}{3}\Rightarrow t\times t=12\times 3\Rightarrow t^2=36\Rightarrow t=6$

$t\times t=12\times 3$

$t^{2}=36$

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Question

The length of the shorter diagonal of a rhombus is 40% that of the longer diagonal. The area of the rhombus is . Give the length of the longer diagonal in terms of .

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Answer

Let be the length of the longer diagonal. Then the shorter diagonal has length 40% of this. Since 40% is equal to $\frac{40}{100 } = \frac{40 \div 20 }{100 \div 20 } = \frac{2}{5}$ , 40% of is equal to $\frac{2}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up, and solve for , in the equation:

$\frac{1}{2} \cdot \frac{2}{5}D \cdot D = K$

$\frac{1}{5}D^{2}= K$

$5\cdot \frac{1}{5}D^{2}=5\cdot K$

$D^{2}=5K$

$D =\sqrt{5K}$

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Question

The length of the shorter diagonal of a rhombus is two-thirds that of the longer diagonal. The area of the rhombus is square yards. Give the length of the longer diagonal, in inches, in terms of .

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Answer

Let be the length of the longer diagonal in yards. Then the shorter diagonal has length two-thirds of this, or $\frac{2}{3}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up the following equation and solve for :

$\frac{1}{2} \cdot \frac{2}{3}D \cdot D = Q$

$\frac{1}{3}D ^{2} = Q$

$3\cdot \frac{1}{3}D ^{2} = 3 \cdot Q$

$D ^{2} = 3Q$

$D = \sqrt{3Q}$

To convert yards to inches, multiply by 36:

$\sqrt{ 3Q } \times 36 = 36\sqrt{ 3Q }$

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Question

The longer diagonal of a rhombus is 20% longer than the shorter diagonal; the rhombus has area . Give the length of the shorter diagonal in terms of .

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Answer

Let be the length of the shorter diagonal. If the longer diagonal is 20% longer, then it measures 120% of the length of the shorter diagonal; this is

$\frac{120}{100} = \frac{120 \div 20}{100 \div 20} = \frac{6}{5}$

of , or $\frac{6}{5}D$ .

The area of a rhombus is half the product of the lengths of its diagonals, so we can set up an equation and solve for :

$\frac{1}{2} \cdot \frac{6}{5}D \cdot D = N$

$\frac{3}{5}D^{2} = N$

$\frac{5}{3} \cdot \frac{3}{5}D^{2} = \frac{5}{3} \cdot N$

$D^{2} = \frac{5N}{3}$

$D =\sqrt{ \frac{5N}{3}}$

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Question

Which of the following shapes is NOT a quadrilateral?

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Answer

A quadrilateral is any two-dimensional shape with sides. The only shape listed that does not have sides is a triangle.

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Question

What other shape can a parallelogram be classified as?

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Answer

A parallelogram can not be classified as a square because a square has to have equal sides, but a parallelogram can have two different side lengths, as long as the opposite side lengths are equal.

A parallelogram can not be classified as a rectangle because a rectangle has to have $90^{\circ}$ angles, and a parallelogram does not.

A parallelogram can not be classified as a triangle because a parallelogram has to have sides and a triangle only has sides.

A parallelogram can not be classified as a rhombus because a rhombus has to have equal sides and a parallelogram does not.

A parallelogram has four sides, and all shapes with four sides are quadrilaterals.

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