### All Basic Geometry Resources

## Example Questions

### Example Question #11 : Radius

If a circle has a circumference of 16π, what would its area be if its radius were halved?

**Possible Answers:**

64π

8π

4π

16π

**Correct answer:**

16π

The circumference of a circle = πd where d = diameter. Therefore, this circle’s diameter must equal 16. Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8. Halving the radius would give us a new radius of 4. To find the area of this new circle, use the formula A=πr² where r = radius. Plug in 4 for r. Area will equal 16π.

### Example Question #12 : Radius

A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.

**Possible Answers:**

361.5

351.5

341.5

346.5

356.5

**Correct answer:**

361.5

The area of the circle is (30/2)^{2}*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5

### Example Question #13 : Radius

The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?

**Possible Answers:**

160π

50π

49π

64π

54π

**Correct answer:**

64π

The original radius would be 4, making the new radius 8 and by the area of a circle (A=π(r)^{2}) the new area would be 64π.

### Example Question #1 : How To Find The Area Of A Circle

A circle with a diameter of 6” sits inside a circle with a radius of 8”. What is the area of the interstitial space between the two circles?

**Possible Answers:**

7π in2

25π in2

28π in2

72π in2

55π in2

**Correct answer:**

55π in2

The area of a circle is πr^{2}.

The diameter of the first circle = 6” so radius of the first circle = 3” so the area = π * 3^{2} = 9π in^{2}

The radius of the second circle = 8” so the area = π * 8^{2} = 64π in^{2}

The area of the interstitial space = area of the first circle – area of the second circle.

Area = 64π in^{2 }- 9π in^{2} = 55π in^{2}

### Example Question #11 : Basic Geometry

If the radius of a circle is tripled, and the new area is 144π what was the diameter of the original circle?

**Possible Answers:**

8

7

4

12

6

**Correct answer:**

8

The area of a circle is A=πr^{2}. Since the radius was tripled 144π =π(3r)^{2}. Divide by π and then take the square root of both sides of the equal sign to get 12=3r, and then r=4. The diameter (d) is equal to twice the radius so d= 2(4) = 8.

### Example Question #12 : Basic Geometry

If the radius of Circle A is three times the radius of Circle B, what is the ratio of the area of Circle A to the area of Circle B?

**Possible Answers:**

15

6

3

12

9

**Correct answer:**

9

We know that the equation for the area of a circle is π r^{2}. To solve this problem, we pick radii for Circles A and B, making sure that Circle A’s radius is three times Circle B’s radius, as the problem specifies. Then we will divide the resulting areas of the two circles. For example, if we say that Circle A has radius 6 and Circle B has radius 2, then the ratio of the area of Circle A to B is: (π 6^{2})/(π 2^{2}) = 36π/4π. From here, the π's cancel out, leaving 36/4 = 9.

### Example Question #11 : How To Find The Area Of A Circle

- A circle is inscribed inside a 10 by 10 square. What is the area of the circle?

**Possible Answers:**

40π

100π

25π

50π

10π

**Correct answer:**

25π

Area of a circle = A = πr^{2}

R = 1/2d = ½(10) = 5

A = 5^{2}π = 25π

### Example Question #61 : Circles

A square has an area of 1089 in^{2}. If a circle is inscribed within the square, what is its area?

**Possible Answers:**

33π in^{2}

1089π in^{2}

16.5 in^{2}

272.25π in^{2}

33 in^{2}

**Correct answer:**

272.25π in^{2}

The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:

A = s^{2} or 1089 = s^{2}. Taking the square root of both sides, we get: s = 33.

Now, based on this, we know that 2r = 33 or r = 16.5. The area of the circle is πr^{2} or π16.5^{2} = 272.25π.

### Example Question #11 : How To Find The Area Of A Circle

A square has an area of 32 in^{2}. If a circle is inscribed within the square, what is its area?

**Possible Answers:**

32π in^{2}

2√2 in^{2}

16π in^{2}

8π in^{2}

4√2 in^{2}

**Correct answer:**

8π in^{2}

The diameter of the circle is the length of a side of the square. Therefore, first solve for the length of the square's sides. The area of the square is:

A = s^{2} or 32 = s^{2}. Taking the square root of both sides, we get: s = √32 = √(2^{5}) = 4√2.

Now, based on this, we know that 2r = 4√2 or r = 2√2. The area of the circle is πr^{2} or π(2√2)^{2} = 4 * 2π = 8π.

### Example Question #26 : Radius

A manufacturer makes wooden circles out of square blocks of wood. If the wood costs $0.25 per square inch, what is the minimum waste cost possible for cutting a circle with a radius of 44 in.?

**Possible Answers:**

7744 – 1936π dollars

1936 dollars

1936 – 484π dollars

1936π dollars

5808 dollars

**Correct answer:**

1936 – 484π dollars

The smallest block from which a circle could be made would be a square that perfectly matches the diameter of the given circle. (This is presuming we have perfectly calibrated equipment.) Such a square would have dimensions equal to the diameter of the circle, meaning it would have sides of 88 inches for our problem. Its total area would be 88 * 88 or 7744 in^{2}.

Now, the waste amount would be the "corners" remaining after the circle was cut. The area of the circle is πr^{2} or π * 44^{2} = 1936π in^{2}. Therefore, the area remaining would be 7744 – 1936π. The cost of the waste would be 0.25 * (7744 – 1936π). This is not an option for our answers, so let us simplify a bit. We can factor out a common 4 from our subtraction. This would give us: 0.25 * 4 * (1936 – 484π). Since 0.25 is equal to 1/4, 0.25 * 4 = 1. Therefore, our final answer is: 1936 – 484π dollars.

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