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# A circle is circumscribed around a square such that all the

tagged by: Brent@GMATPrepNow

#### A circle is circumscribed around a square such that all the

A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the area of the circle, if area of the square is 100 units?

A. 131

B. 137

C. 157

D. 161

E. 174

I'm confused how to set up the formulas here. Can any experts help?

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ardz24 wrote:
A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the area of the circle, if area of the square is 100 units?

A. 131

B. 137

C. 157

D. 161

E. 174

I'm confused how to set up the formulas here. Can any experts help?
Given the are of the square is 100 units. So, the length of one side is 10 units.

The length of diagonal = sqrt(10^2+10^2) = sqrt(200)

As the circle is circumscribed around the square, its diameter is = sqrt(200) and radius = sqrt(200)/2.

Area of circle = Pi * r^2 where r is radius
= 3.1415926 * {sqrt(200)/2}^2 = 3.1415926 * 200 / 4 = 157.07963

Approximate area of circle = 157 square units. Thus, answer is C.

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The question should read "What is the APPROXIMATE area?"

ardz24 wrote:
A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the APPROXIMATE area of the circle, if area of the square is 100 units?

A. 131
B. 137
C. 157
D. 161
E. 174
Here's a diagram of the question:

Since each vertex is a RIGHT ANGLE, we know that the diagonal (shown below) must pass through the center of the circle. In other words, the square's diagonal is also the DIAMETER of the circle.
Also, since the area of square is 100, we know that each side has length 10

Let's let d = the length of the diameter

Since d also represents the HYPOTENUSE of a right triangle with the two legs each having length 10, we can apply the Pythagorean Theorem.
When we do this, we get: 10² + 10² = d²
Simplify: 200 = d²
So, d = √200
Or: d = 10√2

Since the circle's DIAMETER has length 10√2, we can conclude that the circle's RADIUS has length 5√2

What is the APPROXIMATE area of the circle?
= (pi)(5√2
= (pi)(50)

ASIDE: What value of pi is needed here? We COULD use a very precise value of pi, like 3.1416, but that probably isn't necessary.

If we let pi = 3, then the area ≈ (3)(50) ≈ 150. However, given how close the answer choices are, I might want to be a little more precise than this.

If we let pi = 3.1, then the area ≈ (3.1)(50) ≈ 155.
This is very close to answer choice C, so C must be correct.

Cheers,
Brent

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ardz24 wrote:
A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the area of the circle, if area of the square is 100 units?

A. 131

B. 137

C. 157

D. 161

E. 174
Since the area of the square is 100 units, each side is 10.

Note that the diagonal of the square = the diameter of the circle. The diagonal of the square cuts the square into two 45-45-90 triangles, and the ratio of the sides of a 45-45-90 triangle is x : x : x√2. Thus, the diagonal of the square = the diameter of the circle = 10√2, and thus the radius is 5√2.

Therefore, the area of the circle is π(5√2)^2 = 50π ≈ 50 x 3.14 = 157.

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ardz24 wrote:
A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the area of the circle, if area of the square is 100 units?

A. 131

B. 137

C. 157

D. 161

E. 174

I'm confused how to set up the formulas here. Can any experts help?
Hi ardz24,
Let's take a look at your question.

A circle is circumscribed around a square such that all the angles of the squares lie on the circumference.
The area of the square is 100 square units then each of the sides of the square will be 10 units.
It will look like this:

To find the area of the circle we need to find the radius of the circle, which is half the diagonal of the square.
So let's first find the diagonal of the square using Pythagoras theorem.
(Diagonal)^2 = (10)^2 + (10)^2
(Diagonal)^2 = 100 + 100
(Diagonal)^2 = 200
Diagonal = 10√2 units

Now, we can find the radius of the circle,
r = 10√2/2
r = 5√2 units

Area of the circle can now be calculated as:
Area of circle = pi * r^2
Area of circle = 3.14 * (5√2)^2
Area of circle = 3.14 * 25 * 2
Area of circle = 157 units

Therefore, Option C is correct.

I am available if you'd like any followup.

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### GMAT/MBA Expert

GMAT Instructor
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Posted:
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ardz24 wrote:
A circle is circumscribed around a square such that all the angles of the squares lie on circumference. What is the area of the circle, if area of the square is 100 units?

A. 131

B. 137

C. 157

D. 161

E. 174

I'm confused how to set up the formulas here. Can any experts help?
Hi ardz24,
Let's take a look at your question.

A circle is circumscribed around a square such that all the angles of the squares lie on the circumference.
The area of the square is 100 square units then each of the sides of the square will be 10 units.
It will look like this:

To find the area of the circle we need to find the radius of the circle, which is half the diagonal of the square.
So let's first find the diagonal of the square using Pythagoras theorem.
(Diagonal)^2 = (10)^2 + (10)^2
(Diagonal)^2 = 100 + 100
(Diagonal)^2 = 200
Diagonal = 10√2 units

Now, we can find the radius of the circle,
r = 10√2/2
r = 5√2 units

Area of the circle can now be calculated as:
Area of circle = pi * r^2
Area of circle = 3.14 * (5√2)^2
Area of circle = 3.14 * 25 * 2
Area of circle = 157 units

Therefore, Option C is correct.

I am available if you'd like any followup.

_________________
GMAT Prep From The Economist
We offer 70+ point score improvement money back guarantee.
Our average student improves 98 points.

Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now.

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