### All ISEE Upper Level Math Resources

## Example Questions

### Example Question #7 : Radius

What is the area of a circle that has a diameter of inches?

**Possible Answers:**

**Correct answer:**

The formula for finding the area of a circle is . In this formula, represents the radius of the circle. Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius. In order to do this, we divide the diameter by .

Now we use for in our equation.

### Example Question #2 : Area Of A Circle

What is the area of a circle with a diameter equal to 6?

**Possible Answers:**

**Correct answer:**

First, solve for radius:

Then, solve for area:

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

The diameter of a circle is . Give the area of the circle.

**Possible Answers:**

**Correct answer:**

The area of a circle can be calculated using the formula:

,

where is the diameter of the circle, and* * is approximately .

### Example Question #2 : How To Find The Area Of A Circle

The diameter of a circle is . Give the area of the circle in terms of .

**Possible Answers:**

**Correct answer:**

The area of a circle can be calculated using the formula:

,

where is the diameter of the circle and is approximately .

### Example Question #3 : How To Find The Area Of A Circle

The radius of a circle is . Give the area of the circle.

**Possible Answers:**

**Correct answer:**

The area of a circle can be calculated as , where is the radius of the circle, and is approximately .

### Example Question #4 : How To Find The Area Of A Circle

The perpendicular distance from the chord to the center of a circle is , and the chord length is . Give the area of the circle in terms of .

**Possible Answers:**

**Correct answer:**

Chord length = , where is the radius of the circle and is the perpendicular distance from the chord to the circle center.

Chord length =

, where is the radius of the circle and is approximately .

### Example Question #5 : Area Of A Circle

The circumference of a circle is inches. Find the area of the circle.

Let .

**Possible Answers:**

**Correct answer:**

First we need to find the radius of the circle. The circumference of a circle is , where is the radius of the circle.

The area of a circle is where is the radius of the circle.

### Example Question #5 : How To Find The Area Of A Circle

In the above figure, .

What percent of the figure is shaded gray?

**Possible Answers:**

**Correct answer:**

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula :

The outer gray ring is the region between the largest and second-largest circles, and has area

The inner gray ring is the region between the second-smallest and smallest circles, and has area

The total area of the gray regions is

Since out of total area is gray, the percent of the figure that is gray is

.

### Example Question #6 : How To Find The Area Of A Circle

In the above figure, .

Give the ratio of the area of the outer ring to that of the inner circle.

**Possible Answers:**

12 to 1

9 to 1

16 to 1

7 to 1

**Correct answer:**

7 to 1

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for in the formula .

The areas of the largest circle and the second-largest circle are, respectively,

The difference of their areas, which is the area of the outer ring, is

.

The inner circle has area

.

The ratio of these areas is therefore

, or 7 to 1.

### Example Question #7 : How To Find The Area Of A Circle

The above figure depicts a dartboard, in which .

A blindfolded man throws a dart at the target. Disregarding any skill factor and assuming he hits the target, what are the odds against his hitting the white inner circle?

**Possible Answers:**

16 to 1

8 to 1

7 to 1

15 to 1

**Correct answer:**

15 to 1

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The inner and outer circles have radii 1 and 4, respectively, and their areas can be found by substituting each radius for in the formula :

- this is the white inner circle.

The area of the portion of the target outside the white inner circle is , so the odds against hitting the inner circle are

- that is, 15 to 1 odds against.

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