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Basic Geometry : Squares

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Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Side in the square below has a length of 12. What is the length of the diagonal ?

Square_diagonal

Possible Answers:

Cannot be determined from information given.

$12\sqrt{2}$

288

Correct answer:

$12\sqrt{2}$

Explanation:

Diagonal forms a triangle with adjacent sides . Since this is a square we know this is a right triangle and we can use the Pythagorean Theorem to determine the length of . Sides of length form each of the legs and is the hypotenuse. So the equation looks like this:

$12^{2}+12^{2}=d^{2}$

Solve for

$144+144=d^{2}$

$288=d^{2}$

$d=\sqrt{288}$

We can simplify this to

$\sqrt{144\cdot 2}=\sqrt{144}\cdot \sqrt{2}=12\sqrt{2}$

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Example Question #2 : How To Find The Length Of The Diagonal Of A Square

A man wants to build a not-quite-regulation softball field on his property and finds that he only has enough room to make the distance between home plate and first base 44 feet. How far (nearest foot) will it be from home plate to second base, assuming he builds it to that specification?

(Note: the four bases are the vertices of a perfect square, with the bases called home plate, first base, second base, third base, in that order).

Possible Answers:

$88\; \textrm{ft}$

$62\; \textrm{ft}$

$66\; \textrm{ft}$

$76\; \textrm{ft}$

$72\; \textrm{ft}$

Correct answer:

$62\; \textrm{ft}$

Explanation:

The path from home plate to first base is a side of a perfect square; the path from home plate to second base is a diagonal. As two sides and a diagonal form a $45^{\circ } -45^{\circ } -90^{\circ }$ triangle, the diagonal measures $\sqrt{2}$ as long as a side.

The distance to second base from home is $\sqrt{2}$ times the distance to first base:

$d = 44 \cdot \sqrt{2} \approx 44 \cdot 1.414 \approx 62$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Square

A square lot has an area of 1,200 square meters. To the nearest meter, how far is it from one corner to the opposite corner?

Possible Answers:

$300\ meters$

$25\ meters$

$98\ meters$

$49\ meters$

$196\ meters$

Correct answer:

$49\ meters$

Explanation:

A square is also a rhombus, so its area can be calculated as one half the product of its diagonals:

$A= \frac{1 }{2}d^{2}$ ,

where is the common diagonal length.

Since A=1200 , $1200 = \frac{1 }{2}d^{2}$ .

$2400 = d^{2}$

$d = \sqrt{2400} \approx 49$

The distance between opposite corners is about 49 meters.

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Example Question #3 : How To Find The Length Of The Diagonal Of A Square

Find the length of the square's diagonal.

Square_8

Possible Answers:

None of the other answers are correct.

$8\sqrt2$

$4\sqrt5$

Correct answer:

$8\sqrt2$

Explanation:

The diagonal line cuts the square into two equal triangles. Their hypotenuse is the diagonal of the square, so we can solve for the hypotenuse.

We need to use the Pythagorean Theorem: c^2=a^2+b^2 , where a and b are the legs and c is the hypotenuse.

The two legs have lengths of 8. Plug this in and solve for c:

c^2=8^2+8^2=64+64=128

$c=\sqrt{128}=\sqrt{2\cdot 64}=8\sqrt2$

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Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Find the length of the diagonal of a square that has side lengths of cm.

Possible Answers:

$25\text{ cm}$

$4\sqrt{2}$ $\text{cm}$

$4\text{ cm}$

$10\text{ cm}$

$16\text{ cm}$

Correct answer:

$4\sqrt{2}$ $\text{cm}$

Explanation:

You can do this problem in two different ways that lead to the final answer:

1. Pythagorean Theorem

2. Special Triangles (45-45-90)

1. For the first idea, use the Pythagorean Theorem: a^2 +b^2 =c^2 , where a and b are the side lengths of the square and c is the length of the diagonal.

4^2+4^2=c^2

c^2=32

$c=\sqrt{32}$

$c=\sqrt{16*2}=\sqrt{16}\, \times \sqrt{2} = 4\sqrt{2}$

2. If you know that ALL squares can be made into two special right triangles such that their angles are 45-45-90, then there's a formula you could use:

Let's say that your side length of the square is "a". Then the diagonal of the square (or the hypotenuse of the right triangle) will be $a\sqrt{2}$ .

So using this with a=4:

$4\sqrt{2}$

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Example Question #3 : How To Find The Length Of The Diagonal Of A Square

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

Possible Answers:

$24\sqrt{2}$

$2\sqrt{12}$

$12\sqrt{2}$

$48\sqrt{2}$

$\sqrt{12}$

Correct answer:

$12\sqrt{2}$

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:

side^2 + side^2 = hypotenuse^2

12^2+12^2=hypotenuse^2

144 + 144=hypotenuse^2

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

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

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Example Question #5 : How To Find The Length Of The Diagonal Of A Square

The perimeter of a square is units. How many units long is the diagonal of the square?

Possible Answers:

$14\sqrt2$

$7\sqrt2$

$14\sqrt3$

Correct answer:

$7\sqrt2$

Explanation:

From the perimeter, we can find the length of each side of the square. The side lengths of a square are equal by definition therefore, the perimeter can be rewritten as,

4s=28

s=7

Then we use the Pythagorean Theorme to find the diagonal, which is the hypotenuse of a right triangle with each leg being a side of the square.

a^2+b^2=c^2

7^2+7^2=c^2

$c=\sqrt{49+49}=\sqrt{98}=\sqrt{2\cdot49}=7\sqrt2$

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Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Find the length of the diagonal of the square with a side length of .

Possible Answers:

4.5

$8\sqrt2$

$4\sqrt2$

Correct answer:

$4\sqrt2$

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 the diagonal.

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

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

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

For the square given in the question,

$\text{Diagonal}=4\sqrt2$

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Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Find the length of the diagonal of a square with side lengths of .

Possible Answers:

$20\sqrt2$

$5\sqrt2$

$2\sqrt2$

$10\sqrt2$

Correct answer:

$10\sqrt2$

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 the diagonal.

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

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

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

For the square given in the question,

$\text{Diagonal}=10\sqrt2$

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Example Question #4 : How To Find The Length Of The Diagonal Of A Square

Find the length of the diagonal of a square with side lengths of .

Possible Answers:

$15\sqrt2$

$7\sqrt2$

$30\sqrt2$

Correct answer:

$15\sqrt2$

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 the diagonal.

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

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

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

For the square given in the question,

$\text{Diagonal}=15\sqrt2$

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Basic Geometry : Squares

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Example Question #3 : How To Find The Length Of The Diagonal Of A Square

Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Example Question #3 : How To Find The Length Of The Diagonal Of A Square

Example Question #5 : How To Find The Length Of The Diagonal Of A Square

Example Question #1 : How To Find The Length Of The Diagonal Of A Square

Example Question #2 : How To Find The Length Of The Diagonal Of A Square

Example Question #4 : How To Find The Length Of The Diagonal Of A Square

All Basic Geometry Resources

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