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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

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All Basic Geometry Resources

9 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #42 : How To Find The Area Of A Square

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

Possible Answers:

$\frac{1}{2}$

Correct answer:

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.

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

Simplify.

$\text{Side length}=1$

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

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

For this square in question,

$\text{Area}= 1 ^2$

Solve.

$\text{Area}= 1$

Report an Error

Example Question #47 : How To Find The Area Of A Square

If the diagonal of a square is $36\sqrt2$ , what is the area of the square?

Possible Answers:

1672

996

1296

2592

Correct answer:

1296

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.

$\text{Side length}=\frac{36\sqrt2}{\sqrt2}$

Simplify.

$\text{Side length}=36$

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

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

For this square in question,

$\text{Area}= 36 ^2$

Solve.

$\text{Area}= 1296$

Report an Error

Example Question #48 : How To Find The Area Of A Square

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

Possible Answers:

108

Correct answer:

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.

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

Bring the radical to the top of the fraction by multiplying it by one in the form of $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{Side length}=\frac{12}{\sqrt2}\times\frac{\sqrt{2}}{\sqrt{2}}$

Simplify.

$\text{Side length}=\frac{12\sqrt2}{2}$

Reduce.

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

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

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

For this square in question,

$\text{Area}= (6\sqrt2) ^2$

Solve.

$\text{Area}= 72$

Report an Error

Example Question #49 : How To Find The Area Of A Square

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

Possible Answers:

289

361

411

Correct answer:

361

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.

$\text{Area }=361$

Report an Error

Example Question #50 : How To Find The Area Of A Square

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

Possible Answers:

Correct answer:

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.

$\text{side length}=\frac{4}{\sqrt2}$

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{side length}=\frac{4}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Simplify.

$\text{side length}=\frac{4\sqrt2}{2}$

Reduce.

$\text{side length}=2\sqrt2$

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

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

For the square in question,

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

Solve.

$\text{Area }=8$

Report an Error

Example Question #51 : How To Find The Area Of A Square

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

Possible Answers:

Correct answer:

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.

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

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{side length}=\frac{6}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Simplify.

$\text{side length}=\frac{6\sqrt2}{2}$

Reduce.

$\text{side length}=3\sqrt{2}$

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

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

For the square in question,

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

Solve.

$\text{Area }=18$

Report an Error

Example Question #52 : How To Find The Area Of A Square

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

Possible Answers:

Correct answer:

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.

$\text{side length}=\frac{8}{\sqrt2}$

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{side length}=\frac{8}{\sqrt2}\times\frac{\sqrt{2}}\sqrt{2}{}$

Simplify.

$\text{side length}=\frac{8\sqrt2}{2}$

Reduce.

$\text{side length}=4\sqrt2$

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

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

For the square in question,

$\text{Area }=(4\sqrt2) ^2$

Solve.

$\text{Area }=32$

Report an Error

Example Question #81 : Squares

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

Possible Answers:

100

Correct answer:

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.

$\text{side length}=\frac{10}{\sqrt2}$

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{side length}=\frac{10}{\sqrt2}\times\frac{\sqrt{2}}\sqrt{2}{}$

Simplify.

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

Reduce.

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

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

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

For the square in question,

$\text{Area }=(5\sqrt2) ^2$

Solve.

$\text{Area }=50$

Report an Error

Example Question #82 : Squares

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

Possible Answers:

Correct answer:

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.

$\text{side length}=\frac{12}{\sqrt2}$

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}{}$ .

$\text{side length}=\frac{12}{\sqrt2}\times\frac{\sqrt{2}}\sqrt{2}{}$

Simplify.

$\text{side length}=\frac{12\sqrt2}{2}$

Reduce.

$\text{side length}=6\sqrt2$

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

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

For the square in question,

$\text{Area }=(6\sqrt2) ^2$

Solve.

$\text{Area }=72$

Report an Error

Example Question #83 : Squares

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

Possible Answers:

105

Correct answer:

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.

$\text{side length}=\frac{14}{\sqrt2}$

Multiply the fraction by one in the form: $\frac{\sqrt{2}}\sqrt{2}$ .

$\text{side length}=\frac{14}{\sqrt2}\times\frac{\sqrt{2}}\sqrt{2}$

Simplify.

$\text{side length}=\frac{14\sqrt2}{2}$

Reduce.

$\text{side length}=7\sqrt2$

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

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

For the square in question,

$\text{Area }=(7\sqrt2) ^2$

Solve.

$\text{Area }=98$

Report an Error

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All Basic Geometry Resources

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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #42 : How To Find The Area Of A Square

Example Question #47 : How To Find The Area Of A Square

Example Question #48 : How To Find The Area Of A Square

Example Question #49 : How To Find The Area Of A Square

Example Question #50 : How To Find The Area Of A Square

Example Question #51 : How To Find The Area Of A Square

Example Question #52 : How To Find The Area Of A Square

Example Question #81 : Squares

Example Question #82 : Squares

Example Question #83 : Squares

All Basic Geometry Resources

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