Circles - ACT Math

The formula for the area of a circle is A = _πr_2.

We are given the area, and by substitution we know that 13_π_ = _πr_2.

We divide out the π and are left with 13 = _r_2.

We take the square root of r to find that r = √13.

We find the circumference of the circle with the formula C = 2_πr_.

We then plug in our values to find C = 2√13_π_.

First you must draw the diagram. The diagonal of the rectangle is also the diameter of the circle. The diagonal is the hypotenuse of a multiple of 2 of a 3,4,5 triangle, and therefore is 10.
Circumference = π * diameter = 10_π_.

In order to solve this question, first calculate the length of each side of the room.

The length of each side of the room is also equal to the length of the diameter of the largest circular rug that can fit in the room. Since , the circumference is simply

The shape of the garden consists of a rectangle and two semi-circles. Since they are building a fence we need to find the perimeter. The perimeter of the length of the rectangle is 24. The perimeter or circumference of the circle can be found using the equation C=2π(r), where r= the radius of the circle. Since we have two semi-circles we can find the circumference of one whole circle with a radius of 4, which would be 8π.

The formula for the area of a circle is πr2. For this particular circle, the area is 81π, so 81π = πr2. Divide both sides by π and we are left with r2=81. Take the square root of both sides to find r=9. The formula for the circumference of the circle is 2πr = 2π(9) = 18π. The correct answer is 18π.

If the radius is 18 inches, the diameter is 3 feet. The circumference of the tire is therefore 3π by C=d(π). After 200 revolutions, the tire and car have gone 3π x 200 = 600π feet.

The radius is 3. Yielding a circumference of .

We first must calculate the distance between these two points. Recall that the distance formula is:√((x2 - x1)2 + (y2 - y1)2)

For us, it is therefore: √((4 - 2)2 + (6 - 5)2) = √((2)2 + (1)2) = √(4 + 1) = √5

If d = √5, the circumference of our circle is πd, or π√5.

To find the circumference given the radius, simply double the radius and mulitply by .

Thus:

To find the circumference of a circle we use the formula

.

In order to solve we must use the given area to find the radius. Area of a circle has a formula of

.

So we manipulate that formula to solve for the radius.

.

Then we plug in our given area.

.

Now we plug our radius into the circumference equation to get the final answer.

.

To find the circumference of a circle we use the formula

.

In order to solve we must use the given area to find the radius. Area of a circle has a formula of

.

So we manipulate that formula to solve for the radius.

.

Then we plug in our given area.

.

Now we plug our radius into the circumference equation to get the final answer.

.

The questions really asks us to find the perimeter of the track.This would be the two long sides of the rectangle plus the circumferences of each of the two semicircles.

Each long side of the triangle is .

Each semicircle has a circumference that is because the whole circumference is

thus the semicircle circumference would be half of that or,

and .

Since the diameter of the semicircle is equal to the width of the rectangle, . That means each semicircle has a circumference of . Since we have two semicircles on the track, we can sum the two semicircle circumferences and the two long sides of the rectangle that comprise the track to get .

To solve, simply use the formula for the circumference of a circle.

In this particular case the radius of 6 should be substituted into the following equation to solve for the circumference.

Thus,

A question like this is very easy. You merely need to find out what is of the total degrees in a circle. This is:

. That is it!

You know that the area of a circle is computed by the equation:

For our data, this is:

or

Now, the sector is a percentage of the circle. For the areas, this can be represented as the fraction:

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

.

You know that the circumference of a circle is computed by the equation:

For our data, this is:

Now, the sector is a percentage of the circle. For the lengths of the circumference and the arc length, this can be represented as the fraction:

The total degree measure of a circle is, of course, degrees. This means that the sector contains:

.

Do not overthink this question! All you need to remember is that a given circle contains degrees. This means that the sector is merely a percentage of . For our question, this percentage is , which is the same as . So, to calculate, you merely need to multiply:

This is the degree measure of the sector.

Remember that the total degree measure of a circle is . This means that if you have parts into which you have divided your circle, each spoke must be or apart.

Since we know that segment AC is a diameter, this means that the length of the arc ABC must be 180 degrees. This means that the length of the arc AB must be 80 degrees.

Since angle ADB is an inscribed angle, its measure is equal to half of the measure of the angle of the arc that it intercepts. This means that the measure of the angle is half of 80 degrees, or 40 degrees.