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Basic Geometry : How to find the perimeter of a square

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Example Question #31 : How To Find The Perimeter Of A Square

If the diagonal of a square is , what is the perimeter of the square?

Possible Answers:

$15\sqrt2$

$20\sqrt2$

$10\sqrt2$

Correct answer:

$20\sqrt2$

Explanation:

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

Use the Pythagorean Theorem to find the length of one side of the square.

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

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

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

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

For the square given in the question,

$\text{Side length}=\frac{10}{\sqrt{2}}=\frac{10\sqrt2}{2}=5\sqrt2$

Multiply the top and bottom of the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$

$\text{Side length}=\frac{10}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{Side length}=\frac{10\sqrt2}{2}$

Simplify.

$\text{Side length}=5\sqrt2$

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

$\text{Perimeter}=4\times\text{side length}$

For the square in question,

$\text{Perimeter}=4\times 5\sqrt2$

Solve.

$\text{Perimeter}=20\sqrt2$

Report an Error

Example Question #32 : How To Find The Perimeter Of A Square

If the diagonal of a square is , what is the perimeter of the square?

Possible Answers:

$48\sqrt2$

$24\sqrt2$

Correct answer:

$48\sqrt2$

Explanation:

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

Use the Pythagorean Theorem to find the length of one side of the square.

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

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

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

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

For the square given in the question,

$\text{Side length}=\frac{24}{\sqrt{2}}$

Multiply the top and bottom of the fraction by one in the form of $\frac{\sqrt{2}}{\sqrt{2}}$

$\text{Side length}=\frac{24}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{Side length}=\frac{24\sqrt2}{2}$

Simplify.

$\text{Side length}=12\sqrt2$

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

$\text{Perimeter}=4\times\text{side length}$

For the square in question,

$\text{Perimeter}=4\times 12\sqrt2$

Solve.

$\text{Perimeter}=48\sqrt2$

Report an Error

Example Question #461 : Quadrilaterals

Find the perimeter of a square given area is .

Possible Answers:

Correct answer:

Explanation:

To find perimeter, we must first find the side length and then multiply is by .

$A=s^2\Rightarrow s=\sqrt{A}=\sqrt{49}=7$

$P=4s=4\cdot7=28$

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Example Question #32 : How To Find The Perimeter Of A Square

Find the area of the square.

Possible Answers:

900

1600

1200

800

Correct answer:

800

Explanation:

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.

$\text{Area}=\frac{40^2}{2}$

Solve.

$\text{Area}=800$

Report an Error

Example Question #33 : How To Find The Perimeter Of A Square

A square has an area of 49in^2 . Find the perimeter.

Possible Answers:

28in

25in

49in

21in

14in

Correct answer:

28in

Explanation:

Since a square has four equal sides, finding the side lengths from the area is quite simple.

Area = Length(Width)

Since all the sides are equal in a square, the length and width are equal, meaning that Area = side ^2 . Plugging in the numbers gives us:

49 = side^2

$\sqrt{}49 = side$

7 = side

Now that we have found the length of each of the sides of the square, we simply add all the sides together, or multiply by 4 (because there are 4 sides in a square).

7(4) = 28in

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Example Question #34 : How To Find The Perimeter Of A Square

Find the perimeter of a square inscribed in a circle that has a diameter of $2\sqrt2$ .

Possible Answers:

$8\sqrt2$

Correct answer:

Explanation:

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

$\text{diameter}=\text{diagonal}$

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

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{2\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=2$

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

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

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

$\text{Perimeter}=4(2)$

Solve.

$\text{Perimeter}=8$

Report an Error

Example Question #35 : How To Find The Perimeter Of A Square

Find the perimeter of a square inscribed in a circle that has a diameter of $3\sqrt2$ .

Possible Answers:

$12\sqrt2$

Correct answer:

Explanation:

$\text{diameter}=\text{diagonal}$

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

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

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

Simplify.

$\text{side}=3$

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

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

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

$\text{Perimeter}=4(3)$

Solve.

$\text{Perimeter}=12$

Report an Error

Example Question #36 : How To Find The Perimeter Of A Square

Find the perimeter of a square inscribed in a circle that has a diameter of $4\sqrt2$ .

Possible Answers:

$16\sqrt2$

Correct answer:

Explanation:

$\text{diameter}=\text{diagonal}$

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

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{4\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=4$

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

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

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

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

Solve.

$\text{Perimeter}=16$

Report an Error

Example Question #37 : How To Find The Perimeter Of A Square

Find the perimeter of a square inscribed in a circle that has a diameter of $5\sqrt2$ .

Possible Answers:

$20\sqrt2$

$25\sqrt2$

Correct answer:

Explanation:

$\text{diameter}=\text{diagonal}$

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

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{5\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=5$

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

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

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

$\text{Perimeter}=4(5)$

Solve.

$\text{Perimeter}=20$

Report an Error

Example Question #38 : How To Find The Perimeter Of A Square

Find the perimeter of a square inscribed in a circle that has a diameter of $7\sqrt2$ .

Possible Answers:

$28\sqrt2$

Correct answer:

Explanation:

$\text{diameter}=\text{diagonal}$

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

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

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

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

$\text{side}=\sqrt{\frac{\text{Diagonal}^2}{2}}=\frac{\text{Diagonal}\sqrt2}{2}$

Now, substitute in the value of the diagonal to find the length of a side of the square.

$\text{side}=\frac{7\sqrt2(\sqrt2)}{2}$

Simplify.

$\text{side}=7$

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

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

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

$\text{Perimeter}=4(7)$

Solve.

$\text{Perimeter}=28$

Report an Error

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Basic Geometry : How to find the perimeter of a square

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #31 : How To Find The Perimeter Of A Square

Example Question #32 : How To Find The Perimeter Of A Square

Example Question #461 : Quadrilaterals

Example Question #32 : How To Find The Perimeter Of A Square

Example Question #33 : How To Find The Perimeter Of A Square

Example Question #34 : How To Find The Perimeter Of A Square

Example Question #35 : How To Find The Perimeter Of A Square

Example Question #36 : How To Find The Perimeter Of A Square

Example Question #37 : How To Find The Perimeter Of A Square

Example Question #38 : How To Find The Perimeter Of A Square

All Basic Geometry Resources

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