PSAT Math : Plane Geometry

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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Example Question #1 : How To Find The Length Of An Arc

Figure not drawn to scale.

In the figure above, circle C has a radius of 18, and the measure of angle ACB is equal to 100°. What is the perimeter of the red shaded region?

Possible Answers:

18 + 10π

36 + 10π

36 + 36π

18 + 36π

36 + 20π

Correct answer:

36 + 10π

Explanation:

The perimeter of any region is the total distance around its boundaries. The perimeter of the shaded region consists of the two straight line segments, AC and BC, as well as the arc AB. In order to find the perimeter of the whole region, we must add the lengths of AC, BC, and the arc AB.

The lengths of AC and BC are both going to be equal to the length of the radius, which is 18. Thus, the perimeter of AC and BC together is 36.

Lastly, we must find the length of arc AB and add it to 36 to get the whole perimeter of the region.

Angle ACB is a central angle, and it intercepts arc AB. The length of AB is going to equal a certain portion of the circumference. This portion will be equal to the ratio of the measure of angle ACB to the measure of the total degrees in the circle. There are 360 degrees in any circle. The ratio of the angle ACB to 360 degrees will be 100/360 = 5/18. Thus, the length of the arc AB will be 5/18 of the circumference of the circle, which equals 2πr, according to the formula for circumference.

length of arc AB = (5/18)(2πr) = (5/18)(2π(18)) = 10π.

Thus, the length of arc AB is 10π.

The total length of the perimeter is thus 36 + 10π.

The answer is 36 + 10π.

Example Question #1 : Sectors

Circle

In the circle above, the angle A in radians is 

What is the length of arc A?

Possible Answers:

Correct answer:

Explanation:

Circumference of a Circle = 

Arc Length

Example Question #1 : Plane Geometry

Circle_120_degrees

If the area of a circle is , then what is the length of the  arc shown in the diagram?

Possible Answers:

Correct answer:

Explanation:

We are given the area of the circle, but we need to find the circumference in order to find the arc length. The equation for the area of a circle is 

Because we know that the area is 36, we can use that equation to find the radius of the circle.

Divide both sides by 

Take the square root of both sides, and see that the radius is 6.

We can now find the circumference of the circle using the formula 

 

Now that we know the circumference, we can set up a proportion. The length of the 120 degree arc is going to be only a portion of the total circumference of the circle. By putting the degree measure over 360 and setting it equal to x over the circumference, we can find exactly how long the arc is.

When you multiply both sides by , you find the solution:

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

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:

9π in2

4.5π in2

144π in2

18π in2

36π in2

Correct answer:

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 : How To Find The Area Of A Sector

Slide1

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

Possible Answers:

256\pi

144

200

5\pi

100\pi

Correct answer:

100\pi

Explanation:

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:

Area(DBE)=\frac{400}{4}\pi=100\pi   

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

Circle

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

Possible Answers:

Correct answer:

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 #1 : How To Find The Area Of A Sector

Square-missing

 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 3\piwhat is the perimeter of sector?

Picture_16

Possible Answers:

1 + \pi

3 + 2\pi

12 + 2\pi

6 + \pi

12 + \pi

Correct answer:

12 + \pi

Explanation:

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 : Circles

Circle_120_degrees

What is the area of the  sector above if the radius of circle is ?

Possible Answers:

Correct answer:

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 

Example Question #1 : Circles

Two pizzas are made to the same dimensions. The only difference is that Pizza 1 is cut into pieces at 30° angles and Pizza 2 is cut at 45° angles. They are sold by the piece, the first for $1.95 per slice and the second for $2.25 per slice. What is the difference in total revenue between Pizza 2 and Pizza 1?

Possible Answers:

$0

$5.40

–$5.40

–$2.70

$2.70

Correct answer:

–$5.40

Explanation:

First, let's calculate how many slices there are per pizza. This is done by dividing 360° by the respective slice degrees:

Pizza 1: 360/30 = 12 slices

Pizza 2: 360/45 = 8 slices

Now, the total amount made per pizza is calculated by multiplying the number of slices by the respective cost per slice:

Pizza 1: 12 * 1.95 = $23.40

Pizza 2: 8 * 2.25 = $18.00

The difference between Pizza 2 and Pizza 1 is thus represented by: 18 – 23.40 = –$5.40

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