Radius - GMAT Quantitative

To begin this problem, we need to recognize that the distance between points L and K is our diameter. Segment LK passes from one point on circle Q through the center, to another point on circle Q. Sounds like a diameter to me! Use distance formula to find the length of LK.

Plug in our points and simplify:

Now, don't be fooled into choosing 13.15. That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

Let be the sidelength of the square is the square and be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

We find the ratio of to - that is, - as follows:

The correct ratio is .

Let be the radius of circle C. Then Circle B has radius equal to twice this, or , and Circle A has radius equal to three times that, or . The circumference of a circle is equal to times the radius, so the circumferences of the three circles are:

A:

B:

C:

These circumferences add up to , so set up and solve the equation:

Using the formula for the circumference of a circle, we can solve for its radius. Plugging in the given value for the circumference, we have:

A close reading of the question reveals that we are given the circumference and asked to find the radius.

Circumference formula:

So,

So 17 feet

The circumference of a circle, in terms of radius, can be found by:

We are told the circumference with respect to so we can easily solve for the radius:

Notice how the cancels out

Solving this problem is rather straightforward, we just need to remember the circumference of circle is found by and in this case, we're given the circumference in terms of .

Notice how the cancel out

To solve, use the formula for the area of a circle, , and solve for .

A arc of a circle is of the circle. Since the length of this arc is , the circumference is this, or

The radius of a circle is its circumference divided by ; therefore, the radius is

A circle can be divided into three congruent arcs that measure

.

If the three (congruent) chords are constructed, the figure will be an equilateral triangle. The figure is below, along with the altitudes of the triangle:

Since , it follows by way of the 30-60-90 Triangle Theorem that

and

The three altitudes of an equilateral triangle split each other into segments that have ratio 2:1. Therefore,

A circle can be divided into congruent arcs that measure

.

If the (congruent) chords are constructed, the figure will be a regular hexagon. The radius of this hexagon will be equal to the length of one side - one chord of the circle; this radius will coincide with the radius of the circle. Therefore, the radius of the circle is the length of chord , or .

A circle can be divided into four congruent arcs that measure

.

If the four (congruent) chords of the arcs are constructed, they will form a square with sides of length . The diagonal of a square has length times that of a side, which will be

A diagonal of the square is also a diameter of the circle; the circle will have radius half this length, or

The area of a circle with radius is .

Let be the radius of the smaller circle. Its area is . The area of the larger circle is . Since the area of the region between the circles is , and is the difference of these areas, we have

The smaller circle has radius .

The area of a circle with radius is .

Let be the radius of the larger circle. Its area is . The area of the smaller circle is . Since the area of the region between the circles is , and is the difference of these areas, we have

The smaller circle has radius .

If a monster truck's wheels have circumference of , what is the distance from the ground to the center of the wheel?

This question is asking us to find the radius of a circle. the distance from the outside of the circle to the center is the radius. We are given the circumference, so use the following formula:

Then, plug in what we know and solve for r

Let be the sidelength of the square is the square and be the radius of the circle. Then since the areas of the circle and the square are equal, we can set up this equation:

We find the ratio of to - that is, - as follows:

The correct ratio is .

Let be the radius of circle C. Then Circle B has radius equal to twice this, or , and Circle A has radius equal to three times that, or . The circumference of a circle is equal to times the radius, so the circumferences of the three circles are:

A:

B:

C:

These circumferences add up to , so set up and solve the equation:

Using the formula for the circumference of a circle, we can solve for its radius. Plugging in the given value for the circumference, we have:

Using the formula for the area of a circle, we can solve for its radius. Plugging in the given value for the area of the circle, we have: