### All ACT Math Resources

## Example Questions

### Example Question #1 : How To Find The Length Of The Diameter

The perimeter of a circle is 36 π. What is the diameter of the circle?

**Possible Answers:**

3

72

18

36

6

**Correct answer:**

36

The perimeter of a circle = 2 πr = πd

Therefore d = 36

### Example Question #1 : How To Find The Length Of The Diameter

Two legs of a right triangle measure 3 and 4, respectively. What is the area of the circle that circumscribes the triangle?

**Possible Answers:**

**Correct answer:**

For the circle to contain all 3 vertices, the hypotenuse must be the diameter of the circle. The hypotenuse, and therefore the diameter, is 5, since this must be a 3-4-5 right triangle.

The equation for the area of a circle is A = πr^{2}.

### Example Question #1 : How To Find The Length Of The Diameter

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

**Possible Answers:**

**Correct answer:**

1. Use the area to find the radius:

2. Use the radius to find the diameter:

### Example Question #521 : Geometry

What is the diameter of a semi-circle that has an area of ?

**Possible Answers:**

**Correct answer:**

To begin, be very careful to note that the question asks about a *semi-circle*—not a complete circle! This means that a complete circle composed out of two of these semi-circles would have an area of . Now, from this, we can use our area formula, which is:

For our data, this is:

Solving for , we get:

This can be simplified to:

The diameter is , which is or .

### Example Question #1 : Diameter

A circle has an area of . What is the diameter of the circle?

**Possible Answers:**

**Correct answer:**

The equation for the area of a circle is , which in this case equals . Therefore, The only thing squared that equals an integer (which is not a perfect root) is that number under a square root. Therefore, . Since diameter is twice the radius,

### Example Question #4 : How To Find The Length Of The Diameter

Find the diameter given the radius is .

**Possible Answers:**

**Correct answer:**

Diameter is simply twice the radius. Therefore, .

### Example Question #5 : How To Find The Length Of The Diameter

Find the length of the diameter of a circle given the area is .

**Possible Answers:**

**Correct answer:**

To solve, simply use the formula for the area of a circle, solve for r, and multiply by 2 to get the diameter. Thus,

### Example Question #6 : How To Find The Length Of The Diameter

In a group of students, it was decided that a pizza would be divided according to its crust size. Every student wanted 3 inches of crust (measured from the outermost point of the pizza). If the pizza in question had a diameter of 14 inches, what percentage of the pizza was wasted by this manner of cutting the pizza? Round to the nearest hundredth.

**Possible Answers:**

**Correct answer:**

What we are looking at is a way of dividing the pizza according to arc lengths of the crust. Thus, we need to know the total circumference first. Since the diameter is , we know that the circumference is . Now, we want to ask how many ways we can divide up the pizza into pieces of inch crust. This is:

or approximately pieces.

What you need to do is take this amount and subtract off . This is the amount of crust that is wasted. You can then merely divide it by the original amount of divisions:

(You do not need to work in exact area or length. These relative values work fine.)

This is about of the pizza that is wasted.

### Example Question #7 : How To Find The Length Of The Diameter

A circle has an area of . What is its diameter?

**Possible Answers:**

**Correct answer:**

To solve a question like this, first remember that the area of a circle is defined as:

For your data, this is:

To solve for , first divide both sides by . Then take the square root of both sides. Thus you get:

The diameter of the circle is just double that:

Rounding to the nearest hundredth, you get .

### Example Question #111 : Circles

Let represent the area of a circle and represent its circumference. Which of the following equations expresses in terms of ?

**Possible Answers:**

_{}

_{}

_{}

_{}

**Correct answer:**

_{}

The formula for the area of a circle is , and the formula for circumference is . If we solve for C in terms of r, we get

.

We can then substitute this value of r into the formula for the area:

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