How to find the area of a sector

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ACT Math › How to find the area of a sector

Questions 1 - 7

On a circle with a radius of $20\textup{ in}$ , what is the area of a sector having an arc length of $3.5\textup{ in}$ ? Round to the nearest hundredth.

Explanation

To solve this, you need to find the percentage of the given arc with respect to the total circumference. The total circumference of your circle is found merely by using:

$C = 2\pi r$ or, for your data $C = 40\pi$

So, your arc is the following percentage of the circle:

$\frac{3.5}{40\pi}$ or approximately

Now, the total area of your circle is:

$A = \pi r^2$ or $A = 400\pi$

Your sector will be merely $400\pi$ times or .

Stan is making some giant circular pies for his friend's wedding. The pies each have a diameter of and will be cut into equal pieces. Stan wants to have three pieces to himself. What is the surface area of the pie that Stan will eat if he eats three pieces?

$\frac{27\pi}{64}} ft^{2}$

$\frac{64\pi}{27} ft^{2}$

$\frac{8\pi}{3} ft^{2}$

$\frac{32\pi}{9} ft^{2}$

$\frac{7\pi}{15}} ft^{2}$

Explanation

To begin with, we need to break down the word problem to figure out exactly what is being asked of us. We are given a diameter and some clues about how much pie Stan wants to eat.

To solve this problem, we will go through a few steps. First, we have to find the area of a whole pie. Then, we have to find the area of one slice of pie. Finally, we have to multiply the area of a slice of pie by , since Stan wants to eat three slices of pie, not just one.

To find the area of a whole pie, we will need to recall the formula for area of a circle:

$A=\pi r^2$

In this case, we aren't given , but we do know , which is equal to . Since , . Substituting for into the equation for an area of a circle, we get:

$A=\pi 3^2=9\pi$

Each of the answer choices includes $\pi$ , so don't change $9\pi$ into a decimal.

Next, to find the area of one slice of pie, we want to multiply the total area of the circle by the fractional area we are interested in.

We are told in the question that one slice is equivalent to $\frac{1}{64}$ of the pie since each pie will be cut into slices. We also know that Stan wants to eat three slices of pie. Therefore, we will need to multiply the total area by $\frac{3}{64}$ .

So, to find the total area of pie that Stan wants to eat, we will perform the following calculation:

$9\pi*\frac{3}{64}=\frac{27\pi}{64}$

So, Stan wants to eat $\frac{27\pi}{64}:ft^2$ of pie.

Find the area of a 90 degree sector of a circle whose radius is 2.

$\pi$

$4\pi$

$\frac{\pi}{2}$

$2\pi$

Explanation

To solve, first find area. Substitute 2 in for the radius.

$a=\pi{r^2}=\pi*2^2=4\pi$

Then, to find the area of a sector, divide 90 by the total degrees in a circle and multiply it by the area.

Thus,

$A=\frac{90}{360}*4\pi=\frac{1}{4}*4\pi=\pi$

Circle

The radius of the circle above is and $\angle A=45^{\circ}$ . What is the area of the shaded section of the circle?

$2\pi$

$4\pi$

$8\pi$

$16\pi$

$\pi$

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

If a circle, , is centered at the origin, and has an area of , what is the area of the sector defined by the lines and the -axis?

$\frac{\pi}{4}$

$\frac{\pi}{16}$

Explanation

Our first step is to find the angle between the lines and the x-axis. Because one of our lines is the x-axis, we can do this by simply taking the inverse-tangent of the slope of . We know the slope of is equal to 1.

$\tan^{-1}(1)=\frac{\pi}{4}$ radians or degrees. There are $2\pi$ radians or degrees in a circle. Dividing our angle by the whole, we see that our angle is $\frac{1}{8}$ of a complete circle.

Therefore, the area of our sector must be $\frac{1}{8}$ of the total area of the circle. $16 \cdot \frac{1}{8}=\frac{16}{8}=2$ .

A pizza recipe requires drops of hot sauce for every $4\textup{ in}^{2}$ of surface area. If these drops are evenly distributed, what is the amount of hot sauce on a piece of pizza that is $12\textup{ in}$ in diameter, cut into pieces? Round to the nearest drop. Assume that the sauce is evenly spread.

Explanation

To start, calculate the total area of the pizza. This is easily done, using the equation for the area of a circle:

$A = \pi r^2$

Be careful, though. Notice that they give you the diameter, so the radius is :

$A = \pi*6^2 = 36\pi$

Now, if this is cut into eight pieces, this would be:

$\frac{36\pi}{8}$ or approximately per piece.

Now, you need to divide this by to find out the drop amount:

$\frac{14.14}{4}=3.535$

For your answer, you will round to .

A group of students wish to cut a circular pizza of diameter $16\textup{ in}$ into equal slices, each with an angle of $40$^{\circ}$$ . What is the total surface area, in square inches, of each slice of pizza? Round to the nearest hundredth.

22.34

89.36

34.14

19.41

25.81

Explanation

A group of students wish to cut a circular pizza of diameter into pieces with an angle of degrees. What is the total surface area of pizza that each student will recieve if every student eats two pieces of pizza?

To solve a question like this, you need to compute the percentage of the circle that will be represented by a cut of degrees. The total equation for this is: