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Basic Geometry : Quadrilaterals

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Example Questions

Example Question #762 : Plane Geometry

Find the area of a square if it has a diagonal of $\sqrt{34}$ $\displaystyle \sqrt{34}$.

Possible Answers:

$17\sqrt3$ $\displaystyle 17\sqrt3$

$\displaystyle 17$

$17\sqrt2$ $\displaystyle 17\sqrt2$

$\frac{17\sqrt2}{2}$ $\displaystyle \frac{17\sqrt2}{2}$

Correct answer:

$\displaystyle 17$

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.

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

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

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

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

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

$\text{Area}=\frac{(\sqrt{34})^2}{2}$ $\displaystyle \text{Area}=\frac{(\sqrt{34})^2}{2}$

Simplify.

$\text{Area}=\frac{34}{2}$ $\displaystyle \text{Area}=\frac{34}{2}$

$\text{Area}=17$ $\displaystyle \text{Area}=17$

Report an Error

Example Question #763 : Plane Geometry

Find the area of a square if it has a diagonal of $12\sqrt3$ $\displaystyle 12\sqrt3$.

Possible Answers:

$108\sqrt2$ $\displaystyle 108\sqrt2$

$216\sqrt3$ $\displaystyle 216\sqrt3$

216 $\displaystyle 216$

$432\sqrt2$ $\displaystyle 432\sqrt2$

Correct answer:

216 $\displaystyle 216$

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.

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

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

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

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

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

$\text{Area}=\frac{(12\sqrt3)^2}{2}$ $\displaystyle \text{Area}=\frac{(12\sqrt3)^2}{2}$

Simplify.

$\text{Area}=\frac{432}{2}$ $\displaystyle \text{Area}=\frac{432}{2}$

$\text{Area}=216$ $\displaystyle \text{Area}=216$

Report an Error

Example Question #361 : Quadrilaterals

Find the area of the square.

Possible Answers:

1764 $\displaystyle 1764$

$84\sqrt2$ $\displaystyle 84\sqrt2$

882 $\displaystyle 882$

$1264\sqrt3$ $\displaystyle 1264\sqrt3$

Correct answer:

882 $\displaystyle 882$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{42^2}{2}$ $\displaystyle \text{Area}=\frac{42^2}{2}$

Solve.

$\text{Area}=882$ $\displaystyle \text{Area}=882$

Report an Error

Example Question #81 : How To Find The Area Of A Square

Find the area of the square.

Possible Answers:

968 $\displaystyle 968$

$224\sqrt3$ $\displaystyle 224\sqrt3$

1936 $\displaystyle 1936$

$968\sqrt2$ $\displaystyle 968\sqrt2$

Correct answer:

968 $\displaystyle 968$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{44^2}{2}$ $\displaystyle \text{Area}=\frac{44^2}{2}$

Solve.

$\text{Area}=968$ $\displaystyle \text{Area}=968$

Report an Error

Example Question #361 : Quadrilaterals

Find the area of the square.

Possible Answers:

$224\sqrt3$ $\displaystyle 224\sqrt3$

968 $\displaystyle 968$

1936 $\displaystyle 1936$

$968\sqrt2$ $\displaystyle 968\sqrt2$

Correct answer:

968 $\displaystyle 968$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{44^2}{2}$ $\displaystyle \text{Area}=\frac{44^2}{2}$

Solve.

$\text{Area}=968$ $\displaystyle \text{Area}=968$

Report an Error

Example Question #772 : Plane Geometry

Find the area of the square.

Possible Answers:

$1152\sqrt2$ $\displaystyle 1152\sqrt2$

$576\sqrt3$ $\displaystyle 576\sqrt3$

1152 $\displaystyle 1152$

2304 $\displaystyle 2304$

Correct answer:

1152 $\displaystyle 1152$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{48^2}{2}$ $\displaystyle \text{Area}=\frac{48^2}{2}$

Solve.

$\text{Area}=1152$ $\displaystyle \text{Area}=1152$

Report an Error

Example Question #81 : How To Find The Area Of A Square

Find the area of the square.

Possible Answers:

1250 $\displaystyle 1250$

$1250\sqrt3$ $\displaystyle 1250\sqrt3$

$2500\sqrt2$ $\displaystyle 2500\sqrt2$

$625\sqrt3$ $\displaystyle 625\sqrt3$

Correct answer:

1250 $\displaystyle 1250$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{50^2}{2}$ $\displaystyle \text{Area}=\frac{50^2}{2}$

Solve.

$\text{Area}=1250$ $\displaystyle \text{Area}=1250$

Report an Error

Example Question #91 : How To Find The Area Of A Square

Find the area of the square.

Possible Answers:

1352 $\displaystyle 1352$

$676\sqrt3$ $\displaystyle 676\sqrt3$

338 $\displaystyle 338$

2704 $\displaystyle 2704$

Correct answer:

1352 $\displaystyle 1352$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{52^2}{2}$ $\displaystyle \text{Area}=\frac{52^2}{2}$

Solve.

$\text{Area}=1352$ $\displaystyle \text{Area}=1352$

Report an Error

Example Question #772 : Basic Geometry

Find the area of the square.

Possible Answers:

1458 $\displaystyle 1458$

729 $\displaystyle 729$

2916 $\displaystyle 2916$

$365\sqrt2$ $\displaystyle 365\sqrt2$

Correct answer:

1458 $\displaystyle 1458$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{54^2}{2}$ $\displaystyle \text{Area}=\frac{54^2}{2}$

Solve.

$\text{Area}=1458$ $\displaystyle \text{Area}=1458$

Report an Error

Example Question #91 : How To Find The Area Of A Square

Find the area of the square.

Possible Answers:

3136 $\displaystyle 3136$

1998 $\displaystyle 1998$

2428 $\displaystyle 2428$

1568 $\displaystyle 1568$

Correct answer:

1568 $\displaystyle 1568$

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.

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

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

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

Recall how to find the area of a square.

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

Now, substitute in the value of the diagonal to find the area of the square.

$\text{Area}=\frac{56^2}{2}$ $\displaystyle \text{Area}=\frac{56^2}{2}$

Solve.

$\text{Area}=1568$ $\displaystyle \text{Area}=1568$

Report an Error

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Basic Geometry : Quadrilaterals

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #762 : Plane Geometry

Example Question #763 : Plane Geometry

Example Question #361 : Quadrilaterals

Example Question #81 : How To Find The Area Of A Square

Example Question #361 : Quadrilaterals

Example Question #772 : Plane Geometry

Example Question #81 : How To Find The Area Of A Square

Example Question #91 : How To Find The Area Of A Square

Example Question #772 : Basic Geometry

Example Question #91 : How To Find The Area Of A Square

All Basic Geometry Resources

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