Circles - GMAT Quantitative Reasoning

The standard form of the area of a circle with radius and center is

Once we get the equation in standard form, we can find radius , and multiply it by to get the circumference.

Complete the squares:

so can be rewritten as follows:

,

so

And

The equation of a circle centered at the origin is

where is the radius of the circle.

In this equation, , so ; this simplifies to

The circumference of a circle is , so substitute :

The equation of a circle centered at the origin is

,

where is the radius of the circle.

In the equation given in the question stem, , so .

The circumference of a circle is , so substitute :

Since are concentric circles, they share a common center, like sections of a bulls-eye target. Since the radius of is less than half the distance from the edge of to the edge of , we must have circle is inside of circle . (It's helpful to draw a picture to see what's going on!)

Now we can find the radius of by adding and , which is And the equation for finding circumfrence is . Plugging in for gives .

We need to figure out the length of fencing needed to surround a circular enclosure, or in other words, the circumference of the circle.

Circumference equation:

Where is our radius, which is in this case. Plug it in and simplify:

And we have our answer!

Using the formula for the circumference of a circle, we can plug in the given value for the radius and calculate our solution:

Megan, a civil engineer, is designing a roundabout for the city of Madison. She knows that the distance from the edge of the roundabout to the center must be 25 meters. Help Megan find the circumference of the roundabout.

We are asked to find circumference. In order to do so, look at the following formula:

Where r is our radius and C is our circumference.

We are indirectly told that our radius is 25 meters, plug it in to get our answer:

The circumference of a circle is found using the following formula:

Set :

An hour hand rotates 360 degrees for every 12 hours, so the hour hand moves .

There are 4.5 hours between 3 PM and 7:30 PM, so the total degree measure is

.

One full rotation of a circle is , so if a sector covers of a circle, its angle will be of . This gives us:

A circle comprises , so a sector comprising of the circle will have an angle that is of .

Therefore:

A circle comprises , so a sector comprising of the circle will have an angle that is of .

Therefore:

The hour hand moves around a circle from 3PM to 6PM. Since there are 12 hours on a clock and the hand is moving through 3 of them, the hand is moving through a sector comprising of the circle because,

.

Since a circle has , the angle of the sector is:

The town of Thomasville organized a search party to look for a missing chicken. The party consisted of groups of people choosing a sector and searching outward from the center of town. Find the angle for the sector of searched by each group if each group chose an equal sized sector, and there were 120 groups.

Begin by dissecting the question and figurign out exactly what they are asking and telling you. It's a bit wordy, but what we are looking for is the measure of the central angle for each of the search-sectors

We are told that there are 120 equal sectors.

We also know that a circle is made up of

So, to find the central angle of each sector, simply do the following calculation:

, so, by the Multiplication Property of Inequality,

.

The degree measure of an arc is twice that of the inscribed angle that intercepts it, so the above can be rewritten as

.

Here we need to recall the total degree measure of a circle. A circle always has exactly degrees.

Knowing this, we need to utilize two other clues to find the degree measure of .

1. Angle measures degrees, because it is made up of line segment , which is a straight line.

2. Angle can be found by using the following equation. Because we are given the fractional value of its area, we can construct a ratio to solve for angle :

So, to find angle , we just need to subtract our other values from :

So, .

Call the length of a side of Square B . Its perimeter is , which is the radius of Circle A.

The area of the circle is ; that of the square is . Therefore, a sector of the circle with area will be of the circle, which is a sector of measure

This is the kind of question we can't get right if we don't know the trick. In a circle, the size of an angle at the center of the circle, formed by two segments intercepting an arc, is twice the size of the angle formed by two lines intercepting the same arc, provided one of these lines is the diameter of the circle. in other words, is twice .

Thus,

We can see that the points devide the of the circle in 5 equal portions.

The final answer is given simply by which is , this is the angle of a slice of a pizza cut in 5 parts if you will!