# PSAT Math : How to find the area of a sector

## Example Questions

### Example Question #1 : Plane Geometry

A circular, 8-slice pizza is placed in a square box that has dimensions four inches larger than the diameter of the pizza. If the box covers a surface area of 256 in2, what is the surface area of one piece of pizza?

9π in2

18π in2

4.5π in2

144π in2

36π in2

4.5π in2

Explanation:

The first thing to do is calculate the dimensions of the pizza box. Based on our data, we know 256 = s2. Solving for s (by taking the square root of both sides), we get 16 = s (or s = 16).

Now, we know that the diameter of the pizza is four inches less than 16 inches. That is, it is 12 inches. Be careful! The area of the circle is given in terms of radius, which is half the diameter, or 6 inches. Therefore, the area of the pizza is π * 62 = 36π in2. If the pizza is 8-slices, one slice is equal to 1/8 of the total pizza or (36π)/8 = 4.5π in2.

### Example Question #1 : Circles If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE?      Explanation:

Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem: Since AB is a radius of B, we can find the area of circle B via: Angle DBE is a right angle, and therefore of the circle so it follows: ### Example Question #1 : How To Find The Area Of A Sector The radius of the circle above is and .  What is the area of the shaded section of the circle?      Explanation:

Area of Circle = πr2 = π42 = 16π

Total degrees in a circle = 360

Therefore 45 degree slice = 45/360 fraction of circle = 1/8

Shaded Area = 1/8 * Total Area = 1/8 * 16π = 2π

### Example Question #2 : How To Find The Area Of A Sector  is a square. The arc from to is a semicircle with a center at the midpoint of .

All units are in feet.

The diagram shows a plot of land.

The cost of summer upkeep is $2.50 per square foot. In dollars, what is the total upkeep cost for the summer? Possible Answers:     Correct answer: Explanation: To solve this, we must begin by finding the area of the diagram, which is the area of the square less the area of the semicircle. The area of the square is straightforward: 30 * 30 = 900 square feet Because each side is 30 feet long, AB + BC + CD = 30. We can substitute BC for AB and CD since all three lengths are the same: BC + BC + BC = 30 3BC = 30 BC = 10 Therefore the diameter of the semicircle is 10 feet, so the radius is 5 feet. The area of the semi-circle is half the area of a circle with radius 5. The area of the full circle is 52π = 25π, so the area of the semi-circle is half of that, or 12.5π. The total area of the plot is the square less the semicircle: 900 - 12.5π square feet The cost of upkeep is therefore 2.5 * (900 – 12.5π) =$(2250 – 31.25π).

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

In the figure, PQ is the arc of a circle with center O. If the area of the sector is what is the perimeter of sector?       Explanation:

First, we figure out what fraction of the circle is contained in sector OPQ: , so the total area of the circle is .

Using the formula for the area of a circle, , we can see that .

We can use this to solve for the circumference of the circle, , or .

Now, OP and OQ are both equal to r, and PQ is equal to of the circumference of the circle, or .

To get the perimeter, we add OP + OQ + PQ, which give us .

### Example Question #1 : How To Find The Area Of A Sector What is the area of the sector above if the radius of circle is ?      Explanation:

To find the area of a sector, first find the area of the whole circle.

The radius of the circle is 9, so which can be reduced to The area of the sector is only a portion of the total area.To find out exactly how large the area is, set up a proportion where one side equals the angle measure over 360 Multiply both sides by 81 and you will solve for x, which equals  