PSAT Math : How to find the area of a sector

Study concepts, example questions & explanations for PSAT Math

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Example Questions

Example Question #1 : Circles

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?

Possible Answers:

144π in2

18π in2

4.5π in2

9π in2

36π in2

Correct answer:

4.5π in2


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 #3 : Sectors


If B is a circle with line AC = 12 and line BC = 16, then what is the area formed by DBE?

Possible Answers:






Correct answer:



Line AB is a radius of Circle B, which can be found using the Pythagorean Theorem:

AB^2=AC^2+BC^2\rightarrow AB=\sqrt{AC^2+BC^2}=\sqrt{16^2+12^2}=\sqrt{400}=20

Since AB is a radius of B, we can find the area of circle B via:

Area=\pi R^2=\pi(20^2)=400\pi

Angle DBE is a right angle, and therefore of the circle so it follows:


Example Question #1 : Sectors


The radius of the circle above is  and .  What is the area of the shaded section of the circle?

Possible Answers:

Correct answer:


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 #5 : Sectors


 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:



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 #261 : Plane Geometry

In the figure, PQ is the arc of a circle with center O. If the area of the sector is 3\piwhat is the perimeter of sector?


Possible Answers:

3 + 2\pi

12 + \pi

12 + 2\pi

1 + \pi

6 + \pi

Correct answer:

12 + \pi


First, we figure out what fraction of the circle is contained in sector OPQ: \frac{30^{\circ}}{360^{\circ}}= \frac{1}{12}, so the total area of the circle is \dpi{100} \small 12\times 3\pi=36 .

Using the formula for the area of a circle, {\pi}r^{2}, we can see that \dpi{100} \small r=6.

We can use this to solve for the circumference of the circle, 2{\pi}r, or 12{\pi}.

Now, OP and OQ are both equal to r, and PQ is equal to \dpi{100} \small \frac{1}{12} of the circumference of the circle, or {\pi}.

To get the perimeter, we add OP + OQ + PQ, which give us 12+{\pi}.

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 ?

Possible Answers:

Correct answer:


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 

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