Geometry - How to find the area of a square

Know that in a Major League Baseball infield the distance between home plate and first base is 90 feet and the infield is a perfect square.

What is the area of a Major League Baseball infield?

$8100\ ft^2$

$90\ ft^2$

$900\ ft^2$

$360\ ft^2$

$180\ ft^2$

Explanation

Because the infield is a square, the distance between each set of bases is 90 feet.

To find the area of a square you multiply the length by the width.

In this case

$90\times90=8100\ ft^2$ .

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

Explanation

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

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

For the given square,

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

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

$140 ft^{2}$

$20ft^{2}$

$40ft^{2}$

$110ft^{2}$

Explanation

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

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

If the diagonal of a square is $19\sqrt2$ , what is the area of the square?

Explanation

The diagonal of a square is also the hypotenuse of a triangle whose legs are the sides of the square.

Thus, from knowing the length of the diagonal, we can use Pythagorean's Theorem to figure out the side lengths of the square.

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

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

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

$\text{side length}=\frac{\text{Diagonal}}{\sqrt2}$

We can now find the side length of the square in question.

$\text{side length}=\frac{19\sqrt2}{\sqrt2}$

Simplify.

$\text{side length}=19$

Now, recall how to find the area of a square:

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

For the square in question,

$\text{Area }=19 ^2$

Solve.

$\text{Area }=361$

In the figure, a square is inscribed in a circle. If the perimeter of the square is , then what is the area of the shaded region?

Explanation

From the figure, you should notice that the diameter of the circle is also the diagonal of the square.

In order to find the area of the shaded region, we will first need to find the area of the circle and the area of the square.

First, let's find the area of the square.

From the given information, we can find the length of a side of the square.

$\text{Perimeter}=4(\text{side})$

$\text{side}=\frac{\text{Perimeter}}{4}$

Substitute in the value of the perimeter to find the length of a side of the square.

$\text{side}=\frac{8}{4}$

Simplify.

$\text{side}=2$

Now recall how to find the area of a square:

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

Substitute in the value of the side of the square to find the area.

$\text{Area of square}=2^2$

Simplify.

$\text{Area of square}=4$

Now, use the Pythagorean theorem to find the length of the diagonal of the square.

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

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

Simplify.

$\text{Diagonal}=\text{side}\sqrt2$

Substitute in the value of the side of the square to find the length of the diagonal.

$\text{Diagonal}=2\sqrt2$

Recall that the diagonal of the square is the same as the diameter of the circle.

$\text{Diameter}=\text{Diagonal}=2\sqrt2$

From the diameter, we can then find the radius of the circle:

$\text{Radius}=\frac{\text{Diameter}}{2}$

$\text{Radius}=\frac{2\sqrt2}{2}$

Simplify.

$\text{Radius}=\sqrt2$

Now, use the radius to find the area of the circle.

$\text{Area of Circle}=\pi\times\text{radius}^2$

$\text{Area of circle}=\pi\times (\sqrt2)^2$

Simplify.

$\text{Area of circle}=2\pi$

To find the area of the shaded region, subtract the area of the square from the area of the circle.

$\text{Area of Shaded Region}=\text{Area of circle}-\text{Area of square}$

$\text{Area of shaded region}=2\pi-4$

Solve.

$\text{Area of shaded region}=2.28$

A square has perimeter 1.

True or false: The area of the square is $\frac{1}{2}$ .

False

True

Explanation

All four sides of a square have the same length, so the common sidelength is one fourth of the perimeter. The perimeter of the given square is 1, so the length of each side is $\frac{1}{4}$ .

The area of a square is equal to the square of the length of a side, so the area of this square is

$A =\left ( \frac{1}{4} \right )^{2} = \frac{1^{2} }{4^{2} } = \frac{1}{16}$ .

A square has diagonals of length 1. True or false: the area of the square is $\frac{1}{2}$ .

True

False

Explanation

Since a square is a rhombus, its area is equal to half the product of the lengths of its diagonals. Each diagonal has length 1, so the area is equal to

$A = \frac{1}{2} \cdot 1 \cdot 1 = \frac{1}{2}$ .

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

$\frac{5}{2}$

$\frac{15}{2}$

$5\sqrt2$

Explanation

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.

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

Find the area of a square if its diagonal is $5\sqrt3$

$\frac{75}{2}$

$\frac{75\sqrt3}{2}$

$75\sqrt2$

$75\sqrt6$

Explanation

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.

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

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

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

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

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

$\text{Area}=\frac{(5\sqrt3)^2}{2}=\frac{75}{2}$

If the diagonal of a square is $6\sqrt2$ , what is the area of the square?

Explanation

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

We can then use the Pythgorean Theorem to write the following equations:

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

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

$\text{Diagonal}=(\text{side length})\sqrt2$

$\text{Side length}=\frac{\text{Diagonal}}{\sqrt2}$

Now, use this formula and substitute using the given values to find the side length of the square.

$\text{Side length}=\frac{6\sqrt2}{\sqrt2}$

Simplify.

$\text{Side length}=6$

Now, recall how to find the area of a square.

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

For this square in question,

$\text{Area}=6^2$

Solve.

$\text{Area}=36$

How to find the area of a square

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