High School Math : How to find the area of a square

Study concepts, example questions & explanations for High School Math

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Example Questions

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

ABCD and EFGH are squares such that the perimeter of ABCD is 3 times that of EFGH. If the area of EFGH is 25, what is the area of ABCD?

Possible Answers:

15

25

5

225

75

Correct answer:

225

Explanation:

Assign variables such that

One side of ABCD = a

and One side of EFGH = e

Note that all sides are the same in a square. Since the perimeter is the sum of all sides, according to the question:

4a = 3 x 4e = 12e or a = 3e

From that area of EFGH is 25,

e x e = 25 so e = 5

Substitute a = 3e so a = 15

We aren’t done. Since we were asked for the area of ABCD, this is a x a = 225.

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

A square has an area of 36. If all sides are doubled in value, what is the new area?

Possible Answers:

144

72

132

108

48

Correct answer:

144

Explanation:

Let S be the original side length. S*S would represent the original area. Doubling the side length would give you 2S*2S, simplifying to 4*(S*S), giving a new area of 4x the original, or 144.

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

If the perimeter of a square is equal to twice its area, what is the length of one of its sides?

Possible Answers:

3

16

8

4

2

Correct answer:

2

Explanation:

Area of a square in terms of each of its sides:

  Area = S x S

Perimeter of a square:

  Perimeter = 4S

So if 'the perimeter of a square is equal to twice its area':

  2 x Area = Perimeter

  2 x [S x S] = [4S]; divide by 2:

  S x S = 2S; divide by S:

  S = 2

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

Freddie is building a square pen for his pig. He plans to buy x feet of fencing to build the pen. This will result in a pen with an area of p square feet. Unfortunately, he only has enough money to buy one third of the planned amount of fencing. Which expression represents the area of the pen he can build with this limited amount of fencing?

Possible Answers:

p/6

p/3

3p

9p

p/9

Correct answer:

p/9

Explanation:

If Freddie uses x feet of fencing makes a square, each side must be x/4 feet long. The area of this square is (x/4)2 = x2/16 = p square feet.

If Freddie uses one third of x feet of fencing makes a square, each side must be x/12 feet long. The area of this square is (x/12)2x2/144 = 1/9(x2/16) = 1/9(p) = p/9 square feet.

Alternate method:

The scale factor between the small perimeter and the larger perimeter = 1 : 3. Since we're comparing area, a two-dimensional measurement, we can square the scale factor and see that the ratio of the areas is 1: 32 = 1 : 9.

Example Question #221 : Geometry

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

Possible Answers:

128\ cm^{2}

512\ cm^{2}

256\ cm^{2}

32\sqrt{2}\ 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}

Example Question #2 : 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:

1

4

2.25

3

5.06

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.

Example Question #2 : 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:

27\ in.^{2}

48\ in.^{2}

81\ in.^{2}

9\ 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}.

Example Question #2 : 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:

12/14 square units

4/7 square units

6/7 square units

1/7 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.

 

 

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

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

Possible Answers:

\dpi{100} \small 144

\dpi{100} \small 100

\dpi{100} \small 88

\dpi{100} \small 81

\dpi{100} \small 121

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

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

Midpointsquare

Given square , 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:

\sqrt{2}

\frac{\sqrt{2}}{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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