Math - How to find the length of a radius

In a large field, a circle with an area of 144_π_ square meters is drawn out. Starting at the center of the circle, a groundskeeper mows in a straight line to the circle's edge. He then turns and mows ¼ of the way around the circle before turning again and mowing another straight line back to the center. What is the length, in meters, of the path the groundskeeper mowed?

24_π_

12 + 6_π_

12 + 36_π_

24 + 6_π_

24 + 36_π_

Explanation

Circles have an area of πr_2, where r is the radius. If this circle has an area of 144_π, then you can solve for the radius:

πr_2 = 144_π

r 2 = 144

r =12

When the groundskeeper goes from the center of the circle to the edge, he's creating a radius, which is 12 meters.

When he travels ¼ of the way around the circle, he's traveling ¼ of the circle's circumference. A circumference is 2_πr_. For this circle, that's 24_π_ meters. One-fourth of that is 6_π_ meters.

Finally, when he goes back to the center, he's creating another radius, which is 12 meters.

In all, that's 12 meters + 6_π_ meters + 12 meters, for a total of 24 + 6_π_ meters.

If the circumference of a circle is $36\pi$ , what is the radius?

$9\pi$

Explanation

The formula for circumference is $C=2\pi r$ .

Plug in our given information.

$C=2\pi r$

$36\pi=2\pi r$

Divide both sides by $2\pi$ .

$\frac{36\pi}{2\pi}=\frac{2\pi}{2\pi} r$

Find the radius of a circle with area $169\pi$ .

Explanation

Since the formula for the area of a triangle is

$A=\pi r^2$

plug in the given area and isolate for . This yields 13.

What is the radius of a circle with a circumference of $20\pi$ ?

$20\pi$

$10\pi$

Explanation

To find the radius of a circle given the circumference we must first know the equation for the circumference of a circle which is

Then we plug in the circumference into the equation yielding

$20\pi=2r\pi$

We then divide each side by $2\pi$ giving us

The answer is .

Two concentric circles have circumferences of 4π and 10π. What is the difference of the radii of the two circles?

Explanation

The circumference of any circle is 2πr, where r is the radius.

Therefore:

The radius of the smaller circle with a circumference of 4π is 2 (from 2πr = 4π).

The radius of the larger circle with a circumference of 10π is 5 (from 2πr = 10π).

The difference of the two radii is 5-2 = 3.

A circle has an area of 36π inches. What is the radius of the circle, in inches?

Explanation

We know that the formula for the area of a circle is π_r_2. Therefore, we must set 36π equal to this formula to solve for the radius of the circle.

36π = π_r_2

36 = _r_2

6 = r

Circle X is divided into 3 sections: A, B, and C. The 3 sections are equal in area. If the area of section C is 12π, what is the radius of the circle?

Act_math_170_02

Circle X

√12

Explanation

Find the total area of the circle, then use the area formula to find the radius.

Area of section A = section B = section C

Area of circle X = A + B + C = 12π+ 12π + 12π = 36π

Area of circle = where r is the radius of the circle

36π = πr2

36 = r2

√36 = r

6 = r

The circumference of a circle is 45 inches. The circle's radius is ____ inches.

Explanation

When you know the circumference of a circle, you can determine its diameter by dividing the circumference by $\pi$ . Then, when you have the diameter, you can determine the radius by dividing the diameter by 2.

In the figure above, rectangle ABCD has a perimeter of 40. If the shaded region is a semicircle with an area of 18π, then what is the area of the unshaded region?

204 – 18π

96 – 36π

96 – 18π

336 – 18π

336 – 36π

Explanation

In order to find the area of the unshaded region, we will need to find the area of the rectangle and then subtract the area of the semicircle. However, to find the area of the rectangle, we will need to find both its length and its width. We can use the circle to find the length of the rectangle, because the length of the rectangle is equal to the diameter of the circle.

First, we can use the formula for the area of a circle in order to find the circle's radius. When we double the radius, we will have the diameter of the circle and, thus, the length of the rectangle. Then, once we have the rectangle's length, we can find its width because we know the rectangle's perimeter.

Area of a circle = πr2

Area of a semicircle = (1/2)πr2 = 18π

Divide both sides by π, then multiply both sides by 2.

r2 = 36

Take the square root.

r = 6.

The radius of the circle is 6, and therefore the diameter is 12. Keep in mind that the diameter of the circle is also equal to the length of the rectangle.

If we call the length of the rectangle l, and we call the width w, we can write the formula for the perimeter as 2l + 2w.

perimeter of rectangle = 2l + 2w

40 = 2(12) + 2w

Subtract 24 from both sides.

16 = 2w

w = 8.

Since the length of the rectangle is 12 and the width is 8, we can now find the area of the rectangle.

Area = l x w = 12(8) = 96.

Finally, to find the area of just the unshaded region, we must subtract the area of the circle, which is 18π, from the area of the rectangle.

area of unshaded region = 96 – 18π.

The answer is 96 – 18π.

Consider a circle centered at the origin with a circumference of $13\pi$ . What is the x value when y = 3? Round your answer to the hundreths place.

5.77

5.8

5.778

10.00

None of the available answers

Explanation

The formula for circumference of a circle is $C=2\pi r$ , so we can solve for r:

$2\pi r=13\pi$

$\frac{2\pi r}{\pi}=\frac{13\pi}{\pi}$

$2r=13$

$r=\frac{13}{2}=6.5$

We now know that the hypotenuse of the right triangle's length is 13.5. We can form a right triangle from the unit circle that fits the Pythagorean theorem as such:

$x^2+y^2=r^2$