ISEE Lower Level Math : Plane Geometry

Study concepts, example questions & explanations for ISEE Lower Level Math

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Example Questions

Example Question #21 : Geometry

What is the area of a parallelogram if the base is \(\displaystyle 8\), the other side is \(\displaystyle 2\), and the height is \(\displaystyle 3\)?

Possible Answers:

\(\displaystyle 24\)

\(\displaystyle 16\)

\(\displaystyle 8\)\(\displaystyle 12\)

\(\displaystyle 18\)

\(\displaystyle 14\)

Correct answer:

\(\displaystyle 24\)

Explanation:

The area of a parallelogram is \(\displaystyle base*height\) so the answer would be \(\displaystyle 8*3=24\).

Example Question #31 : Geometry

Find the area of the given parallelogram:

Capture

 

Possible Answers:

\(\displaystyle 744 mi^2\)

\(\displaystyle 320 mi^2\)

\(\displaystyle 2304 mi^2\)

\(\displaystyle 1728mi^2\)

Correct answer:

\(\displaystyle 1728mi^2\)

Explanation:

Find the area of the given parallelogram:

Capture

To find the area of a parallelogram, simply do the following:

\(\displaystyle A=b*h\)

Where b is the base, and h is the height. Note that the height is the length of the perpendicular line connecting both bases. In this case, our height is 12mi and our base is 144 miles.

\(\displaystyle A=12mi*144mi=1728mi^2\)

Example Question #32 : Geometry

Find the area of the given parallelogram:

Capture

Possible Answers:

\(\displaystyle 2\textup,319\textup{ mi}^2\)

\(\displaystyle 1\textup,215\textup{ mi}^2\)

\(\displaystyle 3\textup,326\textup{ mi}^2\)

\(\displaystyle 2\textup,315\textup{ mi}^2\)

\(\displaystyle 1\textup,728\textup{ mi}^2\)

Correct answer:

\(\displaystyle 1\textup,728\textup{ mi}^2\)

Explanation:

Find the area of the given parallelogram:

Capture

To find the area of a parallelogram, simply use the following formula for area of a parallelogram:

\(\displaystyle A=b*h\)

Where b is the base, and h is the height. Note that the height is the length of the perpendicular line connecting both bases. In this case, our height is 12 mi and our base is 144 mi.

\(\displaystyle A=12mi*144mi=1728mi^2\)

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

Screen_shot_2013-08-20_at_5.07.49_pm

What is the total surface area of the enclosed region?

Possible Answers:

126cm2

212cm2

112cm2

168cm2

114cm2

Correct answer:

114cm2

Explanation:

Screen_shot_2013-08-20_at_5.21.33_pm

First, we must find the missing lengths. Because we know that the length going horizontally across on the bottom is 6 cm and 8 cm, that must mean that the length going across at the top must also equal this sum. 

So, the length of the top must equal

6 + 8 =

14.

We are given one value of the length on the top, 4. To find the missing horizontal length on the top, we must subtract 4 from 14.

14 – 4 = 10.

In order to find the other missing length, we must observe that the greatest vertical length of this figure is 12 cm. Because we are given 4 cm and 5 cm, we must subtract 4 and 5 from 12 to find the other missing length.

12 – 4 – 5 =

3.

Now, let's divide this enclosed region in three separate rectangles.

The rectangle at the top has a length of 4 cm and width of 3 cm.

The middle rectangle has a length of

10 + 4 =

14 cm

and a width of 5 cm. 

The bottom rectangle has a length of 8 cm and width of 4 cm.

If the formula for the area of a rectangle is length x width, we must now calculate the individual areas of each rectangle and add them up.

Area of top rectangle

4 x 3 = 12cm2

Area of middle rectangle

14 x 5 = 70cm2

Area of bottom rectangle

8 x 4 = 32cm2

12cm2 + 32cm2 + 70cm=

114cm2 

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

Mr. Barker is building a rectangular fence. His yard has an area of 24 feet, and the one side of the fence he's already built is 6 feet long. 

What is the length of the other side (the width) of the fence?

Possible Answers:

\(\displaystyle 7\ feet\)

\(\displaystyle 12\ feet\)

\(\displaystyle 2\ feet\)

\(\displaystyle 8\ feet\)

\(\displaystyle 4\ feet\)

Correct answer:

\(\displaystyle 4\ feet\)

Explanation:

The answer is 4 feet, because \(\displaystyle area=width\times length\)

and a rectangle must have 4 sides, with 2 sides of one length and 2 sides of another.

\(\displaystyle 6\times 4=24\) feet

making the other side 4 feet,

Example Question #11 : Plane Geometry

If Bailey is making a quilt for a bed that measures \(\displaystyle 6\) feet wide and \(\displaystyle 8\) feet long, and she wants there to be an extra foot of quilt to hang over each side of the bed, how much material should she buy.

Possible Answers:

\(\displaystyle 82 ft^{2}\)

\(\displaystyle 80 ft^{2 }\)

\(\displaystyle 78 ft^{2}\)

\(\displaystyle 48 ft^{2}\)

\(\displaystyle 63 ft^{2}\)

Correct answer:

\(\displaystyle 80 ft^{2 }\)

Explanation:

In order to determine how much material Bailey needs, we must first find the area of space she needs to cover. Since Bailey would like the quilt to be an extra foot on each side, we must add \(\displaystyle 2\) feet to both the length and width.

Width  \(\displaystyle 6ft +2ft = 8ft\)

Length \(\displaystyle 8ft +2ft = 10ft\)

Now we apply the formula for the area of a rectangle, which is \(\displaystyle L\times W\).

\(\displaystyle 10\cdot 8= 80 ft^{2}\)

Example Question #12 : Plane Geometry

Rectangle

Give the area of the rectangle in the above diagram.

Possible Answers:

\(\displaystyle 29 \textrm{ in}^2\)

\(\displaystyle 99 \textrm{ in}^2\)

\(\displaystyle 116\textrm{ in}^2\)

\(\displaystyle 58 \textrm{ in}^2\)

\(\displaystyle 198 \textrm{ in}^2\)

Correct answer:

\(\displaystyle 198 \textrm{ in}^2\)

Explanation:

Multiply the length by the width to get the area of the rectangle:

\(\displaystyle 18 \times 11 = 198\)

The area of the rectangle is 198 square inches.

Example Question #34 : Geometry

Which could be the dimensions of a rectangle with the area \(\displaystyle 36\textup{ cm}^{2}\)?

Possible Answers:

\(\displaystyle 4 \textup{ cm}\times 8 \textup{ cm}\)

\(\displaystyle 18 \textup{ cm}\times 18 \textup{ cm}\)

\(\displaystyle 2 \textup{ cm}\times 13 \textup{ cm}\)

\(\displaystyle 3 \textup{ cm}\times 12 \textup{ cm}\)

Correct answer:

\(\displaystyle 3 \textup{ cm}\times 12 \textup{ cm}\)

Explanation:

To find the area, simply multiply the sides of the rectangle. The only sides which add up to \(\displaystyle 36\) are:

\(\displaystyle 3 \textup{ cm}\times 12 \textup{ cm}\)

Example Question #6 : How To Find The Area Of A Rectangle

Rectangle

Give the area of the above rectangle in square centimeters.

Possible Answers:

\(\displaystyle 10,000 \textrm{ cm}^{2}\)

\(\displaystyle 50,000 \textrm{ cm}^{2}\)

\(\displaystyle 70,000 \textrm{ cm}^{2}\)

\(\displaystyle 140,000 \textrm{ cm}^{2}\)

\(\displaystyle 100,000 \textrm{ cm}^{2}\)

Correct answer:

\(\displaystyle 100,000 \textrm{ cm}^{2}\)

Explanation:

Since 1 meter = 100 centimeters, multiply each dimension by 100 to convert meters to centimeters. This makes the dimensions 200 centimeters by 500 centimeters. 

Use the area formula, substituting \(\displaystyle L = 500, W = 200\):

\(\displaystyle A = LW\)

\(\displaystyle A = 500 \times 200 = 100,000 \textrm{ cm}^{2}\)

Example Question #35 : Geometry

Mr. Smith is planting a rectangular garden with a length of 5 ft and a width of 7 ft. What is the area of the garden?

Possible Answers:

\(\displaystyle 35ft^2\)

\(\displaystyle 24ft^2\)

\(\displaystyle 20ft^2\)

\(\displaystyle 28ft^2\)

Correct answer:

\(\displaystyle 35ft^2\)

Explanation:

The area of a rectangle can be calculated by multiplying the length by the width.

\(\displaystyle A=l\times w=5\: ft\times7\: ft=35\: ft^2\)

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