How to find the area of a square - Geometry

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Question

A square is circumscribed on a circle with a 6 inch radius. What is the area of the square, in square inches?

Answer

We know that the radius of the circle is also half the length of the side of the square; therefore, we also know that the length of each side of the square is 12 inches.

We need to square this number to find the area of the square.

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Question

A square garden has sides that are feet long. In square feet, what is the area of the garden?

Answer

Use the following formula to find the area of a square:

$\text{Area}=(\text{side length})^2$

For the given square,

$\text{Area}=8^2=64$

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Question

In square meters, find the area of a square that has a side length of meter.

Answer

Use the following formula to find the area of a square:

$\text{Area}=(\text{side length})^2$

For the given square,

$\text{Area}=1^2=1$

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Question

Jennifer wants to put down carpet on her bedroom floor that is a square with side lengths of feet. In square feet, how much carpet is needed?

Answer

Use the following formula to find the area of a square:

$\text{Area}=(\text{side length})^2$

For the given square,

$\text{Area}=14^2=196$

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Question

Find the area of the square.

Answer

The diagonal of a square is also the hypotenuse of a right isosceles triangle that has the sides of the square as its legs.

Thus, we can use the Pythagorean Theorem to find the length of the sides of the square.

$\text{Diagonal}^2=\text{side}^2+\text{side}^2$

$\text{Diagonal}^2=2(\text{side}^2)$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

Recall how to find the area of a square.

$\text{Area}=\text{side}^2=\frac{\text{Diagonal}^2}{2}$

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{58^2}{2}$

Solve.

$\text{Area}=1682$

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Question

Find the area of a square given side length of 4.

Answer

To find the area of a square multiply the width with the length. In the case of a square, all sides are the same length thus, the area is simply the side length squarred.

To solve, simply use the formula for the area of a square and let,

.

Thus,

.

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Question

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.

Answer

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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Question

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

Answer

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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Question

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

Answer

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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Question

Find the area of a square if it has a diagonal of $3\sqrt2$ .

Answer

The diagonal of a square is also the hypotenuse of a triangle.

Recall how to find the area of a square:

$\text{Area}=\text{side}^2$

Now, use the Pythagorean theorem to find the area of the square.

$\text{side}^2+\text{side}^2=\text{Diagonal}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{Area}=\text{side}^2=\frac{\text{Diagonal}^2}{2}$

Substitute in the length of the diagonal to find the area of the square.

$\text{Area}=\frac{(3\sqrt2)^2}{2}$

Simplify.

$\text{Area}=\frac{18}{2}$

$\text{Area}=9$

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Question

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?

Answer

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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Question

The sides of a square garden are 10 feet long. What is the area of the garden?

Answer

The formula for the area of a square is

$Area = s^{2}$

where is the length of the sides. So the solution can be found by

$Area=(10)^{2}=100 ft^{2}$

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Question

Eric has 160 feet of fence for a parking lot he manages. If he is using all of the fencing, what is the area of the lot assuming it is square?

Answer

The area of a square is equal to its length times its width, so we need to figure out how long each side of the parking lot is. Since a square has four sides we calculate each side by dividing its perimeter by four.

$\frac{160}{4}=40$

Each side of the square lot will use 40 feet of fence.

$Area=length \times width$

$40\ ft\times 40\ ft=1600\ ft^{2}$ .

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Question

Side in the square shown below has a length of 13 meters. What is the total area of the square?

Answer

Since the shape in question is a square, we know that is the length of all four sides.

The formula for the area of a square is $area=base\times height$ .

In this case, area = $a\times a$ , or $a^{2}$ . Plug in the given value of a, 13 meters, to solve for the area:

$13^{2}=169$

Remember to use the correct units, in this square meters.

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Question

Find the area of a square if it has a diagonal of $\sqrt5$ .

Answer

The diagonal of a square is also the hypotenuse of a triangle.

Recall how to find the area of a square:

$\text{Area}=\text{side}^2$

Now, use the Pythagorean theorem to find the area of the square.

$\text{side}^2+\text{side}^2=\text{Diagonal}^2$

$2(\text{side})^2=\text{Diagonal}^2$

$\text{side}^2=\frac{\text{Diagonal}^2}{2}$

$\text{Area}=\text{side}^2=\frac{\text{Diagonal}^2}{2}$

Substitute in the length of the diagonal to find the area of the square.

$\text{Area}=\frac{(\sqrt5)^2}{2}$

Simplify.

$\text{Area}=\frac{5}{2}$

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Question

The length of line in the figure below is 15 inches. What is the area of the square ?

Answer

Since is a square, the side is the same length as all of the other three sides.

We get area by multiplying length by width (or base by height, if you prefer), since all sides are equal, it looks like this:

$15\times15=225$

Don't forget, the units are SQUARE INCHES.

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Question

Point A is the center of the circle.

Figure ABCD is a square.

Segments AB and AD are radii of the circle.

The radius of the circle is units.

Find the area of the green-colored shape.

Answer

Square ABCD contains both the red and green shapes. The red shape is equal to the area of one-fourth of the circle. Finding the area of square ABCD and subtracting only the area of the red shape will give the area of only the green shape.

Since ABCD is a square, angle BAC is a right angle that sits at the center of the circle (point A). Since a right angle is 90o and a circle is 360o, the red shape's area must be one quarter (or $\frac{1}{4}$ ) of the entire circle's area. Use the equation $A = \pi r^2$ to find the area of the entire circle, then multiply this by $\frac{1}{4}$ to find the area of only the red shape.

$A_{red} = (\pi (4)^2) * (\frac{1}{4})$

$A_{red} = 4\pi$

Subtracting this from the area of the square gives the area of the green area outside of the circle.

$A_{square}=s^2=4^2=16$

$A_{green}=A_{square}-A_{red}$

$A_{green}=16-4\pi$

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Question

What is the area of the following square if the length of is $2\sqrt{2}$ ?

Answer

We need to find the length of the side of the square in order to find the area. The diagonal makes two $45^{\circ}-45^{\circ}-90^{\circ}$ triangles with the sides of the square. Using the $45^{\circ}-45^{\circ}-90^{\circ}$ special triangle ratio of $n-n-n\sqrt{2}$ , we know that if the hypotenuse is $2\sqrt{2}$ then the length of each side must be . The area of the square is $side^{2} = 2^{^{2}} = 4$ .

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Question

One side of a square is 6 inches long. What is the area of the square in inches?

Answer

To find the area of a square, you only need to know one side. The length of one side squared is the area.

$6^{2}=36$

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Question

Find the area of a square that has side lengths of mm.

Answer

The area of any square is: , so

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