### All ACT Math Resources

## Example Questions

### Example Question #31 : How To Find The Area Of A Circle

What is the area of a cirlce with a circumference of ?

**Possible Answers:**

**Correct answer:**

The formula for the circumference of a circle is . Because we are given the circumference we substitute for and solve for , yielding .

Next, we need to plug in our value for into the formula for the area of a circle.

and get

### Example Question #31 : Radius

A circle has a cricumference of . Given this information, find the area of the circle.

**Possible Answers:**

**Correct answer:**

To find the area of a circle, we use the formula

where r is the radius.

However, the problem does not give us the circles radius. In order to solve for the area we must find the radius using the circumference. Circumference of a circle follows the equation

,

so since we know the circumference we can manipulate the equation and plug in our value to solve for radius.

.

Now that we know the radius we simply plug it into the area formula to solve for our final answer.

### Example Question #31 : How To Find The Area Of A Circle

In the following diagram, the radius is given. What is area of the shaded region?

**Possible Answers:**

** **

**Correct answer:**

** **

This question asks you to apply the concept of area in finding both the area of a circle and square. Since the cirlce is inscribed in the square, we know that its diameter (two times the radius) is the same length as one side of the square. Since we are given the radius, , we can find the area of both the circle and square.

Square:

^{ }

This gives us the area for the entire square.

The bottom half of the square has area .

Now that we have this value, we must find the area that the circle occupies. The area of a circle is given by .

So the area of this circle will be .

The bottom half of the circle has half that area:

Now that we have both our values, we can subtract the bottom half of the circle from the bottom half of the square to give us the shaded region:

### Example Question #31 : Radius

What is the area of a circle, in terms of , that has a radius of 6 inches?

**Possible Answers:**

**Correct answer:**

To find the area of a circle with a given radius, use the formula:

### Example Question #32 : How To Find The Area Of A Circle

Find the area of a circle given the radius is 8.

**Possible Answers:**

**Correct answer:**

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

Since the question gives us the value of the radius, we can substitute 8 in for the radius to solve for the area.

Thus,

### Example Question #33 : How To Find The Area Of A Circle

Find the area of a circle given a radius of 1.

**Possible Answers:**

**Correct answer:**

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

In this particular case, substitute one in for the radius in the following equation.

Thus,

### Example Question #571 : Act Math

A circle has a maximum chord length of inches. What is its area in square inches?

**Possible Answers:**

**Correct answer:**

We can use the fact that the longest chord of a circle is its diameter.

In other words, this circle has a diameter of inches, meaning a radius of inches.

From there, finding the area is straightforward:

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

If a circle has circumference , what is its area?

**Possible Answers:**

**Correct answer:**

If the circumference is , then since we know . We further know that , so

### Example Question #1 : How To Find Circumference

If a circle has an area of , what is the circumference of the circle?

**Possible Answers:**

**Correct answer:**

The formula for the area of a circle is πr^{2}. For this particular circle, the area is 81π, so 81π = πr^{2}. Divide both sides by π and we are left with r^{2}=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.

### Example Question #1 : How To Find Circumference

A circle with an area of 13*π* in^{2} is centered at point *C*. What is the circumference of this circle?

**Possible Answers:**

√13*π*

√26*π*

26*π*

2√13*π*

13*π*

**Correct answer:**

2√13*π*

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*.