Squares

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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.

$\text{Area }=361$

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.

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

The perimeter of a square is 48. What is the length of its diagonal?

$12\sqrt{2}$

$\sqrt{12}$

$24\sqrt{2}$

$48\sqrt{2}$

$2\sqrt{12}$

Explanation

Perimeter = side * 4

48 = side * 4

Side = 12

We can break up the square into two equal right triangles. The diagonal of the sqaure is then the hypotenuse of these two triangles.

Therefore, we can use the Pythagorean Theorem to solve for the diagonal:

$\sqrt{144+144}=\sqrt{hypotenuse^2}$

$hypotenuse=\sqrt{2\times 144}=\sqrt{2\times 12\times 12}=12\sqrt{2}$

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.

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

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,

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

What is the length of a rectangular room with a perimeter of and a width of

Explanation

We have the perimeter and the width, so we can plug those values into our equation and solve for our unknown.

Subtract from both sides

Divide by both sides

$\frac{50}{2}=\frac{2l}{2}$

What is the perimeter of a square with area 196 square inches?

$56 ; \textrm{in}$

$28 ; \textrm{in}$

$112 ; \textrm{in}$

$64; \textrm{in}$

It cannot be determined from the information given.

Explanation

A square with area 196 square inches has sidelength $\sqrt{196} = 14$ inches, and therefore has perimeter $4 \cdot 14 = 56$ inches

Angela has a garden that she wants to put a fence around. How much fencing will she need if her garden is by

Explanation

The fence is going around the garden, so this is a perimeter problem.

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.

$\text{Area}=36$

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