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SSAT Upper Level Math : Geometry

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Example Questions

Example Question #5 : How To Find The Area Of A Rectangle

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.

Possible Answers:

$12\ inches$

$8\ inches$

$20\ inches$

$10\ inches$

$16\ inches$

Correct answer:

$20\ inches$

Explanation:

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 10+10=20 inches.

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

A rectangle has the width of t-1 and the diagonal length of t+1 . Give the area of the rectangle in terms of .

Possible Answers:

$2\sqrt{t}(t-1)$

$4\sqrt{t}(t-1)$

$\sqrt{t}(t-1)$

$\sqrt{t}(t+1)$

$2\sqrt{t}(t+1)$

Correct answer:

$2\sqrt{t}(t-1)$

Explanation:

First we need to find the height of the rectangle. Since the width and the diagonal lengths are known, we can use the Pythagorean Theorem to find the height of the rectangle:

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

$t+1=\sqrt{w^{2}+(t-1)^{2}}$

$\left ( t+1 \right )^{2}=w^{2}+\left ( t-1 \right )^{2}$

So we have:

$w^2=(t+1)^2-(t-1)^{2}$

$\Rightarrow w^2=(t^2+2t+1)-(t^2-2t+1)=4t$

$\Rightarrow w=\sqrt{4t}=2\sqrt{t}$

So we can get:

$Area=wh=2\sqrt{t}(t-1)$

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Example Question #3 : How To Find The Area Of A Rectangle

The perimeter of a rectangle is 800 inches. The width of the rectangle is 60% of its length. What is the area of the rectangle?

Possible Answers:

$37,500 \textrm{ in}^{2}$

$13,500\textrm{ in}^{2}$

$50,000\textrm{ in}^{2}$

$15,000\textrm{ in}^{2}$

$75,000\textrm{ in}^{2}$

Correct answer:

$37,500 \textrm{ in}^{2}$

Explanation:

Let be the length of the rectangle. Then its width is 60% of this, or 0.60 L . The perimeter is the sum of the lengths of its sides, or

L + 0.60 L + L + 0.60 = 3.20 L ; we set this equal to 800 inches and solve for :

3.20 L = 800

$3.20 L \div 3.20 = 800 \div 3.20$

L = 250

The width is therefore

$0.60 \times 250 = 150$ .

The product of the length and width is the area:

$250 \times 150 = 37,500$ square inches.

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Example Question #4 : How To Find The Area Of A Rectangle

Rectangle A has length 40 inches and height 24 inches. Rectangle B has length 30 inches and height 28 inches. Rectangle C has length 72 inches, and its area is the mean of the areas of the other two rectangles. What is the height of Rectangle C?

Possible Answers:

$18\textrm{ in}$

$15\textrm{ in}$

$13\frac{1}{3}\textrm{ in}$

$12\frac{1}{2}\textrm{ in}$

$10\textrm{ in}$

Correct answer:

$12\frac{1}{2}\textrm{ in}$

Explanation:

The area of a rectangle is the product of the length and its height, Rectangle A has area $40 \times 24 = 960$ square inches; Rectangle B has area $30 \times 28 = 840$ square inches.

The area of Rectangle C is the mean of these areas, or

$\frac{960+840}{2} = 900$ square inches, so its height is this area divided by its length:

$900 \div 72 = 12 \frac{1}{2}$ inches.

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Example Question #9 : How To Find The Area Of A Rectangle

The area of a rectangle is square feet. The width of the rectangle is four-sevenths of its length. Give the length of the rectangle in inches in terms of .

Possible Answers:

$\frac{7K} {48}$

$\frac{ \sqrt{ 7K } }{24}$

$\frac{ \sqrt{ 7K } }{2}$

$6\sqrt{ 7K }$

21K

Correct answer:

$6\sqrt{ 7K }$

Explanation:

Let be the length in feet. Then the width of the rectangle in feet is four-sevenths of this, or $\frac{4}{7}L$ . The area is equal to the product of the length and the width, so set up this equation and solve for :

$\frac{4}{7}L \cdot L = K$

$\frac{4}{7}L ^{2}= K$

$\frac{7} {4}\cdot \frac{4}{7}L ^{2}= \frac{7} {4}\cdot K$

$L ^{2}= \frac{7K} {4}$

$L =\sqrt{ \frac{7K} {4}} =\frac{ \sqrt{ 7K } }{\sqrt{4}}=\frac{ \sqrt{ 7K } }{2}$

Since this is the length in feet, we multiply this by 12 to get the length in inches:

$\frac{ \sqrt{ 7K } }{2} \cdot 12 = 6\sqrt{ 7K }$

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Example Question #11 : How To Find The Area Of A Rectangle

Rectangle A has length 40 inches and height inches; Rectangle B has length 30 inches and height inches; Rectangle C has height inches, and its area is the sum of those of the other two rectangles. What is its length?

Possible Answers:

$22\frac{1}{2} \textrm{ in}$

$\frac{70H}{3} \textrm{ in}$

$\frac{45H}{2} \textrm{ in}$

$23\frac{1}{3} \textrm{ in}$

The correct answer is not among the other choices.

Correct answer:

The correct answer is not among the other choices.

Explanation:

The area of a rectangle is the product of the length and its height.

Rectangle A has area 40H square inches, and Rectangle B has an area of $30 \cdot 2H = 60 H$ square inches, so the sum of their areas is 40H + 60 H = 100 H square inches. This is the area of Rectangle C; divide it by height inches to get a length of

$\frac{100 H}{4H} = \frac{100}{4} = 25$ inches. This answer is not among the given choices.

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Example Question #12 : How To Find The Area Of A Rectangle

The perimeter of a rectangle is 490 centimeters. The width of the rectangle is three-fourths of its length. What is the area of the rectangle?

Possible Answers:

$19,600\textrm{ cm}^{2}$

$58,800 \textrm{ cm}^{2}$

$11,025 \textrm{ cm}^{2}$

$14,700 \textrm{ cm}^{2}$

$29,400 \textrm{ cm}^{2}$

Correct answer:

$14,700 \textrm{ cm}^{2}$

Explanation:

Let be the length of the rectangle. Then its width is three-fourths of this, or $\frac{3}{4} L$ . The perimeter is the sum of the lengths of its sides, or

$L + \frac{3}{4} L + L + \frac{3}{4} L = \frac{7}{2} L$ .

Set this equal to 490 centimeters and solve for :

$\frac{7}{2} L = 490$

$\frac{2} {7} \cdot \frac{7}{2} L =\frac{2} {7} \cdot 490$

L = 140

The length of the rectangle is 140 centimeters; the width is three-fourths of this, or

$\frac{3}{4} \times 140 = 105$ centimeters.

The area is the product of the length and the width:

$140 \times 105 = 14,700$ square centimeters.

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Example Question #13 : How To Find The Area Of A Rectangle

A basketball team wants to paint a 4-foot wide border around its court to make sure fans don't get too close to the action. If the court is 94 by 50 feet, and one can of paint can cover 300 square feet, how many cans of paint does the team need to ensure that the entire border is painted?

(Assume that you cannot buy partial cans of paint.)

Possible Answers:

Correct answer:

Explanation:

We begin this problem by finding the difference of two areas: the larger rectangle bounded by the outer edge of the border and the smaller rectangle that is the court itself.

The larger rectangle is 5916 square feet $\left(94+4+4\right\times\left(50+4+4 \right )$ , and the court is 4700 square feet $(94\times 50)$ .

The difference, 1296 , repesents the area of the border.

Now we divide this by 300 , which is just a bit over . But since we can't leave square feet unpainted, we have to round up to cans of paint.

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Example Question #11 : How To Find The Area Of A Rectangle

Box

The above diagram shows a rectangular solid. The shaded side is a square. Give the total surface area of the solid.

Possible Answers:

1,000

800

500

2,000

Correct answer:

1,000

Explanation:

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

Box 2

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

Top, bottom, front, back (four surfaces): $20 \times 10 = 200$

Left, right (two surfaces): $10 \times 10 = 100$

The total area: $4 \times 200 + 2 \times 100 = 800 + 200 = 1,000$

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Example Question #82 : Areas And Perimeters Of Polygons

Box

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

Possible Answers:

400x

$2x^{2} + 80x$

80x + 800

$20 x^{2}$

Correct answer:

$2x^{2} + 80x$

Explanation:

Since a square has four sides of equal length, the solid looks like this:

Box

The areas of each of the individual surfaces, each of which is a rectangle, are the product of their dimensions:

Front, back, top, bottom (four surfaces): 20 x

Left, right (two surfaces): $x \cdot x = x^{2}$

The total surface area is therefore

$2 \cdot x^{2} + 20x \cdot 4 = 2 x^{2} + 80x$

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SSAT Upper Level Math : Geometry

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #5 : How To Find The Area Of A Rectangle

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

Example Question #3 : How To Find The Area Of A Rectangle

Example Question #4 : How To Find The Area Of A Rectangle

Example Question #9 : How To Find The Area Of A Rectangle

Example Question #11 : How To Find The Area Of A Rectangle

Example Question #12 : How To Find The Area Of A Rectangle

Example Question #13 : How To Find The Area Of A Rectangle

Example Question #11 : How To Find The Area Of A Rectangle

Example Question #82 : Areas And Perimeters Of Polygons

All SSAT Upper Level Math Resources

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