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High School Math : Squares

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

If the diagonal of a square measures $16\sqrt{2} \ cm$ , what is the area of the square?

Possible Answers:

$32\sqrt{2}\ cm^{2}$

$128\ cm^{2}$

$512\ cm^{2}$

$256\ cm^{2}$

$64\sqrt{2}\ cm^{2}$

Correct answer:

$256\ cm^{2}$

Explanation:

This is an isosceles right triangle, so the diagonal must equal $\sqrt{2}$ times the length of a side. Thus, one side of the square measures $16\ cm$ , and the area is equal to $(16 \ cm)^{2} = 256\ cm^{2}$

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

A square $A$ has side lengths of $z$ . A second square $B$ has side lengths of $2.25z$ . How many $A's$ can you fit in a single $B$ ?

Possible Answers:

$2.25$

$3$

$5.06$

$4$

$1$

Correct answer:

$5.06$

Explanation:

The area of $A$ is $n$ , the area of $B$ is $5.0625n$ . Therefore, you can fit 5.06 $A's$ in $B$ .

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

The perimeter of a square is $12\ in.$ If the square is enlarged by a factor of three, what is the new area?

Possible Answers:

$9\ in.^{2}$

$27\ in.^{2}$

$48\ in.^{2}$

$81\ in.^{2}$

$36\ in.^{2}$

Correct answer:

$81\ in.^{2}$

Explanation:

The perimeter of a square is given by $P=4s=12$ so the side length of the original square is $3\ in.$ The side of the new square is enlarged by a factor of 3 to give $s=9\ in.$

So the area of the new square is given by $A = s^{2} = (9)^{2} = 81 in^{2}$ .

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

How much more area does a square with a side of 2r have than a circle with a radius r? Approximate π by using 22/7.

Possible Answers:

1/7 square units

6/7 square units

12/14 square units

4/7 square units

Correct answer:

6/7 square units

Explanation:

The area of a circle is given by A = πr² or 22/7r²

The area of a square is given by A = s² or (2r)² = 4r²

Then subtract the area of the circle from the area of the square and get 6/7 square units.

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Example Question #2 : Squares

If the perimeter of a square is 44 centimeters, what is the area of the square in square centimeters?

Possible Answers:

$\dpi{100} \small 81$

$\dpi{100} \small 144$

$\dpi{100} \small 100$

$\dpi{100} \small 121$

$\dpi{100} \small 88$

Correct answer:

$\dpi{100} \small 121$

Explanation:

Since the square's perimeter is 44, then each side is $\dpi{100} \small \frac{44}{4}=11$ .

Then in order to find the area, use the definition that the

$\dpi{100} \small Area=side^{2}$

$\dpi{100} \small 11^{2}=121$

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

Midpointsquare

Given square ABCD , with midpoints on each side connected to form a new, smaller square. How many times bigger is the area of the larger square than the smaller square?

Possible Answers:

$2\sqrt{2}$

$\frac{\sqrt{2}}{2}$

$\sqrt{2}$

Correct answer:

Explanation:

Assume that the length of each midpoint is 1. This means that the length of each side of the large square is 2, so the area of the larger square is 4 square units. $A=s^{2}$

To find the area of the smaller square, first find the length of each side. Because the length of each midpoint is 1, each side of the smaller square is $\sqrt{2}$ (use either the Pythagorean Theorem or notice that these right trianges are isoceles right trianges, so $s, s, s\sqrt{2}$ can be used).

The area then of the smaller square is 2 square units.

Comparing the area of the two squares, the larger square is 2 times larger than the smaller square.

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Example Question #91 : Quadrilaterals

Square_with_diagonal

Find the area of a square with a diagonal of d=2.5cm .

Possible Answers:

$\dpi{100} 4.25cm^{2}$

Not enough information to solve

$6.25cm^{2}$

$5.0cm^{2}$

$3.125cm^{2}$

Correct answer:

$3.125cm^{2}$

Explanation:

A few facts need to be known to solve this problem. Observe that the diagonal of the square cuts it into two right isosceles triangles; therefore, the length of a side of the square to its diagonal is the same as an isosceles right triangle's leg to its hypotenuse: $1:\sqrt{2}$ .

$\frac{s}{d}=\frac{1}{\sqrt{2}}$

$\dpi{100} \frac{s}{2.5cm}=\frac{1}{\sqrt{2}}$

Rearrange an solve for .

$s=\frac{2.5cm}{\sqrt{2}}$

Now, solve for the area using the formula $A=s^{2}$ .

$A=(\frac{2.5cm}{\sqrt{2}})^{2}$

$\dpi{100} A=\frac{6.25cm^{2}}{2}$

$\rightarrow 3.125cm^{2}$

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Example Question #91 : Quadrilaterals

If the ratio of the sides of two squares is 2:1 , what is the ratio of the areas of those two squares?

Possible Answers:

8:1

2:1

4:1

6:1

Correct answer:

4:1

Explanation:

Express the ratio of the two sides of the squares as x:y = 2:1 . The area of each square is one side multiplied by itself, so the ratios of the areas would be x^2:y^2 = 2^2:1^2 . The right side of this equation simplifies to a ratio of 4:1 .

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Example Question #3 : Squares

If a completely fenced-in square-shaped yard requires 140 feet of fence, what is the area, in square feet, of the lot?

Possible Answers:

140

4900

1225

Correct answer:

1225

Explanation:

Since the yard is square in shape, we can divide the perimeter(140ft) by 4, giving us 35ft for each side. We then square 35 to give us the area, 1225 feet.

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Example Question #93 : Quadrilaterals

What is the area of a square with a diagonal of $5\sqrt{2}$ ?

Possible Answers:

100

$25\sqrt{2}$

Correct answer:

Explanation:

The formula for the area of a square is A=s*s . However, the problem gives us a diagonal and not a side.

Remember that all sides of a square are equal, so the diagonal cuts the square into two equal triangles, each a 45:45:90 right triangle.

If we use the Pythagorean Theorem, we see:

s^2+s^2=d^2

2s^2=d^2

Plug in our given diagonal to solve.

$2s^2=(5\sqrt{2})^2$

2s^2=50

s^2=25

$\sqrt{s^2}=\sqrt{25}$

s=5

From here we can plug our answer back into our original equation:

A=s*s

A=5*5

A=25

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High School Math : Squares

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

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

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

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

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

Example Question #2 : Squares

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

Example Question #91 : Quadrilaterals

Example Question #91 : Quadrilaterals

Example Question #3 : Squares

Example Question #93 : Quadrilaterals

All High School Math Resources

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