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Basic Geometry : Squares

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

Example Questions

Example Question #61 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $6\sqrt7$ $\displaystyle 6\sqrt7$.

Possible Answers:

$126\sqrt3$ $\displaystyle 126\sqrt3$

126 $\displaystyle 126$

$252\sqrt2$ $\displaystyle 252\sqrt2$

$63\sqrt3$ $\displaystyle 63\sqrt3$

Correct answer:

126 $\displaystyle 126$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(6\sqrt7)^2}{2}=\frac{252}{2}=126$ $\displaystyle \text{Area}=\frac{(6\sqrt7)^2}{2}=\frac{252}{2}=126$

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Example Question #62 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $7\sqrt5$ $\displaystyle 7\sqrt5$.

Possible Answers:

$46\sqrt5$ $\displaystyle 46\sqrt5$

$\frac{245\sqrt2}{4}$ $\displaystyle \frac{245\sqrt2}{4}$

$\frac{245}{2}$ $\displaystyle \frac{245}{2}$

$71\sqrt3$ $\displaystyle 71\sqrt3$

Correct answer:

$\frac{245}{2}$ $\displaystyle \frac{245}{2}$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(7\sqrt5)^2}{2}=\frac{245}{2}$ $\displaystyle \text{Area}=\frac{(7\sqrt5)^2}{2}=\frac{245}{2}$

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Example Question #63 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $9\sqrt{10}$ $\displaystyle 9\sqrt{10}$.

Possible Answers:

$405\sqrt2$ $\displaystyle 405\sqrt2$

405 $\displaystyle 405$

$\frac{405}{2}$ $\displaystyle \frac{405}{2}$

$405\sqrt3$ $\displaystyle 405\sqrt3$

Correct answer:

405 $\displaystyle 405$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(9\sqrt{10})^2}{2}=\frac{810}{2}=405$ $\displaystyle \text{Area}=\frac{(9\sqrt{10})^2}{2}=\frac{810}{2}=405$

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Example Question #91 : Squares

Find the area of a square if its diagonal is $7\sqrt{10}$ $\displaystyle 7\sqrt{10}$.

Possible Answers:

$\frac{245}{2}$ $\displaystyle \frac{245}{2}$

245 $\displaystyle 245$

$490\sqrt2$ $\displaystyle 490\sqrt2$

$\frac{245\sqrt3}{2}$ $\displaystyle \frac{245\sqrt3}{2}$

Correct answer:

245 $\displaystyle 245$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(7\sqrt{10})^2}{2}=\frac{490}{2}=245$ $\displaystyle \text{Area}=\frac{(7\sqrt{10})^2}{2}=\frac{490}{2}=245$

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Example Question #62 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $6\sqrt{10}$ $\displaystyle 6\sqrt{10}$.

Possible Answers:

$360\sqrt3$ $\displaystyle 360\sqrt3$

$180\sqrt2$ $\displaystyle 180\sqrt2$

180 $\displaystyle 180$

$90\sqrt3$ $\displaystyle 90\sqrt3$

Correct answer:

180 $\displaystyle 180$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(6\sqrt{10})^2}{2}=\frac{360}{2}=180$ $\displaystyle \text{Area}=\frac{(6\sqrt{10})^2}{2}=\frac{360}{2}=180$

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Example Question #62 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $8\sqrt{10}$ $\displaystyle 8\sqrt{10}$.

Possible Answers:

320 $\displaystyle 320$

$320\sqrt3$ $\displaystyle 320\sqrt3$

$320\sqrt2$ $\displaystyle 320\sqrt2$

$160\sqrt3$ $\displaystyle 160\sqrt3$

Correct answer:

320 $\displaystyle 320$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(8\sqrt{10})^2}{2}=\frac{640}{2}=320$ $\displaystyle \text{Area}=\frac{(8\sqrt{10})^2}{2}=\frac{640}{2}=320$

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Example Question #67 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $11\sqrt2$ $\displaystyle 11\sqrt2$.

Possible Answers:

$\frac{121\sqrt2}{2}$ $\displaystyle \frac{121\sqrt2}{2}$

121 $\displaystyle 121$

$\frac{121}{2}$ $\displaystyle \frac{121}{2}$

$121\sqrt3$ $\displaystyle 121\sqrt3$

Correct answer:

121 $\displaystyle 121$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(11\sqrt{2})^2}{2}=\frac{242}{2}=121$ $\displaystyle \text{Area}=\frac{(11\sqrt{2})^2}{2}=\frac{242}{2}=121$

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Example Question #68 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $15\sqrt{10}$ $\displaystyle 15\sqrt{10}$.

Possible Answers:

875 $\displaystyle 875$

1125 $\displaystyle 1125$

775 $\displaystyle 775$

2250 $\displaystyle 2250$

Correct answer:

1125 $\displaystyle 1125$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(15\sqrt{10})^2}{2}=\frac{2250}{2}=1125$ $\displaystyle \text{Area}=\frac{(15\sqrt{10})^2}{2}=\frac{2250}{2}=1125$

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Example Question #71 : How To Find The Area Of A Square

Find the area of a square if its diagonal is $12\sqrt{14}$ $\displaystyle 12\sqrt{14}$.

Possible Answers:

1008 $\displaystyle 1008$

$1008\sqrt2$ $\displaystyle 1008\sqrt2$

$1008\sqrt5$ $\displaystyle 1008\sqrt5$

$504\sqrt{3}$ $\displaystyle 504\sqrt{3}$

Correct answer:

1008 $\displaystyle 1008$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(12\sqrt{14})^2}{2}=\frac{2016}{2}=1008$ $\displaystyle \text{Area}=\frac{(12\sqrt{14})^2}{2}=\frac{2016}{2}=1008$

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Example Question #751 : Plane Geometry

Find the area of a square if its diagonal is $\sqrt{30}$ $\displaystyle \sqrt{30}$.

Possible Answers:

$30\sqrt2$ $\displaystyle 30\sqrt2$

$\displaystyle 15$

$\displaystyle 30$

$15\sqrt2$ $\displaystyle 15\sqrt2$

Correct answer:

$\displaystyle 15$

Explanation:

The diagonal of a square is also the hypotenuse of a $\displaystyle 45-45-90$ triangle.

Recall how to find the area of a square:

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

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

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

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

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

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

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

$\text{Area}=\frac{(\sqrt{30})^2}{2}=\frac{30}{2}=15$ $\displaystyle \text{Area}=\frac{(\sqrt{30})^2}{2}=\frac{30}{2}=15$

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All Basic Geometry Resources

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Basic Geometry : Squares

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #61 : How To Find The Area Of A Square

Example Question #62 : How To Find The Area Of A Square

Example Question #63 : How To Find The Area Of A Square

Example Question #91 : Squares

Example Question #62 : How To Find The Area Of A Square

Example Question #62 : How To Find The Area Of A Square

Example Question #67 : How To Find The Area Of A Square

Example Question #68 : How To Find The Area Of A Square

Example Question #71 : How To Find The Area Of A Square

Example Question #751 : Plane Geometry

All Basic Geometry Resources

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