SSAT Middle Level Math : Plane Geometry

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

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Example Questions

Example Question #14 : How To Find The Perimeter Of A Square

Find the perimeter of the square in the figure.

 Screen_shot_2014-02-26_at_4.37.24_pm

Possible Answers:

\(\displaystyle 62\ \textup{cm}\)

\(\displaystyle 64.5\ \textup{cm}\)

\(\displaystyle 58\ \textup{cm}\)

\(\displaystyle 59.5\ \textup{cm}\)

\(\displaystyle 63\ \textup{cm}\)

Correct answer:

\(\displaystyle 62\ \textup{cm}\)

Explanation:

Finding perimeter is easiest with squares. Perimeter is the distance around the outside of the figure. There are two good ways to find the perimeter. The first and longer method of finding perimeter is simply to add up the length of each side. Thus, we would take the length of top of the square and add it to the length of the right side. We'd continue all the way around. The formula for this method could be written like so: 

top + right side + bottom + left side = perimeter.

Since a square has equal sides all around, we could also write the formulalike so:

\(\displaystyle s+s+s+s=perimeter\)

The second and easier way of finding perimeter for a square is a simplification of the first method. Instead of adding the same number four times we could simply multiply by 4. Written as a formula, this would look like this:

\(\displaystyle 4\ \times \ s=perimeter\)

For the this question, your work should then look like one of these two options:

\(\displaystyle 15.5cm + 15.5cm + 15.5cm + 15.5cm = 62.0 cm\)

or

\(\displaystyle 15.5cm \times 4 = 62.0 cm\)

Example Question #15 : How To Find The Perimeter Of A Square

Find the perimeter of the square in the figure. 

Screen_shot_2014-02-26_at_4.39.27_pm

Possible Answers:

\(\displaystyle 66.8\ \textup{cm}\)

\(\displaystyle 60.5\ \textup{cm}\)

\(\displaystyle 64.8\ \textup{cm}\)

\(\displaystyle 62\ \textup{cm}\)

\(\displaystyle 68.4\ \textup{cm}\)

Correct answer:

\(\displaystyle 64.8\ \textup{cm}\)

Explanation:

Finding perimeter is easiest with squares. Perimeter is the distance around the outside of the figure. There are two good ways to find the perimeter. The first and longer method of finding perimeter is simply to add up the length of each side. Thus, we would take the length of top of the square and add it to the length of the right side. We'd continue all the way around. The formula for this method could be written like so: 

top + right side + bottom + left side = perimeter.

Since a square has equal sides all around, we could also write the formulalike so:

\(\displaystyle s+s+s+s=perimeter\)

The second and easier way of finding perimeter for a square is a simplification of the first method. Instead of adding the same number four times we could simply multiply by 4. Written as a formula, this would look like this:

\(\displaystyle 4\ \times \ s=perimeter\)

For the this question, your work should then look like one of these two options:

\(\displaystyle 16.2cm + 16.2cm + 16.2cm + 16.2cm = 64.8 cm\)

or

\(\displaystyle 16.2cm \times 4 = 64.8 cm\)

Example Question #51 : Plane Geometry

A sqaure has sides measuring \(\displaystyle 5\) inches. What is this square's perimeter?

Possible Answers:

\(\displaystyle 20\: in\)

\(\displaystyle 20\: in^{2}\)

\(\displaystyle 10\: in\)

\(\displaystyle 10\: in^{2}\)

\(\displaystyle 5\: in\)

Correct answer:

\(\displaystyle 20\: in\)

Explanation:

The perimeter of a shape is the distance around the outside boundry of that shape.

In order to find the perimeter, add up the lengths of the sides of that shape.

Because the sides of a square all have the same length, add \(\displaystyle 5+5+5+5\). Equivalently, multiply the side length by the number of sides: \(\displaystyle 4*5=20\).

Because perimeter measures length, the unit of the answer should be inches, not inches2.

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

What is the area of a square with perimeter 64 inches?

Possible Answers:

\(\displaystyle 4,096 \; \textrm{in}^{2}\)

It cannot be determined from the information given.

\(\displaystyle 256 \; \textrm{in}^{2}\)

\(\displaystyle 512 \; \textrm{in}^{2}\)

\(\displaystyle 1,024 \; \textrm{in}^{2}\)

Correct answer:

\(\displaystyle 256 \; \textrm{in}^{2}\)

Explanation:

The perimeter of a square is four times its sidelength, so a square with perimeter 64 inches has sides with length 16 inches. Use the area formula:

\(\displaystyle A = s^{2} = 16^{2} = 256\)

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

The area of the square is 81. What is the sum of the lengths of three sides of the square?

Possible Answers:

\(\displaystyle 81\)

\(\displaystyle 54\)

\(\displaystyle 18\)

\(\displaystyle 27\)

\(\displaystyle 9\)

Correct answer:

\(\displaystyle 27\)

Explanation:

A square that has an area of 81 has sides that are the square root of 81 (side2 = area for a square).  Thus each of the four sides is 9.  The sum of three of these sides is \(\displaystyle 9 + 9 + 9 = 27\).

Example Question #93 : Geometry

Cube

 

What is the total area of the surface of the cube shown in the above diagram?

Possible Answers:

\(\displaystyle 73 \frac{1}{2} \textrm{ in}^{2}\)

\(\displaystyle 62 \frac{1}{2} \textrm{ in}^{2}\)

\(\displaystyle 49 \textrm{ in}^{2}\)

\(\displaystyle 42 \textrm{ in}^{2}\)

\(\displaystyle 42 \frac{7}{8} \textrm{ in}^{2}\)

Correct answer:

\(\displaystyle 73 \frac{1}{2} \textrm{ in}^{2}\)

Explanation:

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

\(\displaystyle 3 \frac{1}{2} \times 3 \frac{1}{2} = \frac{7}{2} \times \frac{7}{2} = \frac{49}{4} \textrm{ in}^{2}\)

Then multiply this by six:

\(\displaystyle 6 \times \frac{49}{4} =\frac{6}{1} \times \frac{49}{4} = \frac{3}{1} \times \frac{49}{2} = \frac{147}{2} = 73 \frac{1}{2} \textrm{ in}^{2}\)

Example Question #91 : Geometry

Cube

 

What is the total area of the surface of the cube shown in the above diagram?

Possible Answers:

\(\displaystyle 105.456 \textrm{ cm}^{2}\)

\(\displaystyle 31.2\textrm{ cm}^{2 }\)

\(\displaystyle 17.576 \textrm{ cm}^{2}\)

\(\displaystyle 40.56 \textrm{ cm}^{2}\)

\(\displaystyle 27.04 \textrm{ cm}^{2}\)

Correct answer:

\(\displaystyle 40.56 \textrm{ cm}^{2}\)

Explanation:

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

\(\displaystyle 2.6 \times 2.6 = 6.76 \textrm{ cm}^{2}\)

Then multiply this by six:

\(\displaystyle 6 \times 6.76 = 40.56 \textrm{ cm}^{2}\)

Example Question #92 : Geometry

Cube

What is the total area of the surface of the cube shown in the above diagram?

 

Possible Answers:

\(\displaystyle 15,625\textrm{ in}^{2}\)

\(\displaystyle 2,500\textrm{ in}^{2}\)

\(\displaystyle 3,750\textrm{ in}^{2}\)

\(\displaystyle 5,000\textrm{ in}^{2}\)

\(\displaystyle 625\textrm{ in}^{2}\)

Correct answer:

\(\displaystyle 3,750\textrm{ in}^{2}\)

Explanation:

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

\(\displaystyle 25 \times 25 = 625 \textrm{ in}^{2}\)

Then multiply this by six:

\(\displaystyle 6 \times 625 = 3,750\textrm{ in}^{2}\)

Example Question #21 : Squares

Cube

What is the total area of the surface of the cube shown in the above diagram?

Possible Answers:

\(\displaystyle 144 \textrm{ cm}^2\)

\(\displaystyle 864 \textrm{ cm}^2\)

\(\displaystyle 288 \textrm{ cm}^2\)

\(\displaystyle 1,728 \textrm{ cm}^2\)

\(\displaystyle 576 \textrm{ cm}^2\)

Correct answer:

\(\displaystyle 864 \textrm{ cm}^2\)

Explanation:

A cube comprises six faces, each of which is a square. To find its total surface area, find the area of one face by squaring its sidelength:

\(\displaystyle 12 \times 12 = 144 \textrm{ cm}^{2}\)

Then multiply this by six:

\(\displaystyle 6 \times 144 = 864\textrm{ cm}^{2}\)

Example Question #4 : Geometry

A square is 9 feet long on each side.  How many smaller squares, each 3 feet on a side can be cut out of the larger square?

Possible Answers:

\(\displaystyle 81\)

\(\displaystyle 27\)

\(\displaystyle 3\)

\(\displaystyle 9\)

\(\displaystyle 6\)

Correct answer:

\(\displaystyle 9\)

Explanation:

Each side can be divided into three 3-foot sections.  This gives a total of \(\displaystyle 3\times3=9\) squares.  Another way of looking at the problem is that the total area of the large square is 81 and each smaller square has an area of 9.  Dividing 81 by 9 gives the correct answer.

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