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High School Math : Plane Geometry

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Example Questions

Example Question #4 : How To Find An Angle In A Polygon

What is the measure of one interior angle of a regular twenty-three-sided polygon (nearest tenth of a degree)?

Possible Answers:

$164.3^{\circ }$

$154.3^{\circ }$

$169.3^{\circ }$

$174.3^{\circ }$

$159.3^{\circ }$

Correct answer:

$164.3^{\circ }$

Explanation:

The measure of each interior angle of a regular polygon with sides is $\frac{180 (N-2)}{N}$ . We can substitute N = 23 to obtain the angle measure:

$\frac{180 (N-2)}{N} = \frac{180 (23-2)}{23} = \frac{180 \cdot 21}{23}\approx 164.3^{\circ }$

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Example Question #5 : How To Find An Angle In A Polygon

A regular polygon has interior angles which measure $162^{\circ }$ each. How many sides does the polygon have?

Possible Answers:

Correct answer:

Explanation:

The easiest way to answer this is to note that, since an interior angle and an exterior angle form a linear pair - and thus, a supplementary pair - each exterior angle would have measure $180-162 = 18^{\circ }$ . Since 360 divided by the number of sides of a regular polygon is equal to the measure of one of its exterior angles, we are seeking such that

$\frac{360}{N} = 18$

Solve for :

$\frac{360}{N}\cdot N= 18\cdot N$

18N = 360

$18N \div 18= 360\div 18$

N = 20

The polygon has 20 sides.

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Example Question #1 : How To Find The Length Of A Side Of A Polygon

If the area of a regular octagon is 160 and the apothem is 8, what is the side length?

Possible Answers:

12.25

Correct answer:

Explanation:

To find the side length from the area of an octagon and the apothem we must use the area of a polygon which is

First plug in our numbers for area and the apothem to get

$160=\frac{1}{2}(8)(perimeter)$

Then multiply to get 160=4(perimeter)

Then divide both sides by 4 to get the perimeter of the figure. $\frac{160}{4}=40$

When we have the perimeter of a regular polygon, to find the side length we must divide by the number of sides of the polygon, in this case 8. $\frac{40}{8}=5$

After dividing we find the side length is

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Example Question #1 : Circles

A sector comprises 20% of a circle. What is the central angle of the sector?

Possible Answers:

Correct answer:

Explanation:

Proporations can be used to solve for the central angle. Let equal the angle of the sector.

$\frac{20}{100}=\frac{x}{360}$

Cross mulitply:

7200=100x

Solve for :

$\frac{7200}{100}=\frac{100x}{100}$

72=x

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Example Question #1 : How To Find The Angle For A Percentage Of A Circle

Arcs

$m \angle AXC = 135^{\circ }$ ; $arc\ \widehat{AB} = 24^{\circ }$ ; $arc\ \widehat{AC} = 94^{\circ }$

Find the degree measure of $arc\ \widehat{CD}$ .

Possible Answers:

$164^{\circ}$

$66^{\circ }$

Not enough information is given to answer this question.

$132^{\circ }$

$82^{\circ}$

Correct answer:

$66^{\circ }$

Explanation:

When two chords of a circle intersect, the measure of the angle they form is half the sum of the measures of the arcs they intercept. Therefore,

$m \angle AXB = \frac{m\; \widehat{AB} + m\; \widehat{CD} }{2}$

Since $\angle AXB$ and $\angle AXC$ form a linear pair, $m \angle AXB + m \angle AXC = 180$ , and $m \angle AXB = 180 - m \angle AXC = 180 - 135 = 45^{\circ }$ .

Substitute $m\; \widehat{AB} = 24$ and $m \angle AXB = 45$ into the first equation:

$45 = \frac{24 + m\; \widehat{CD} }{2}$

$90 =24 + m\; \widehat{CD}$

$66 = m\; \widehat{CD}$

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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:

–$2.70

–$5.40

$5.40

$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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Example Question #3 : Circles

What percentage of a full circle is the following sector? (Round to the nearest tenth of a percent.)

Arch

Note: The figure is not drawn to scale.

Possible Answers:

$14.4\%$

$20\%$

$16.7\%$

$25\%$

$33.3\%$

Correct answer:

$16.7\%$

Explanation:

In order to find the percentage of a sector from an angle, you need to know that a full circle is $360^{\circ}$ .

Therefore, we can find the percentage by dividing the angle of the sector by $360^{\circ}$ and then multiplying by 100:

$Percentage = \frac{\measuredangle}{360^{\circ}}\cdot 100$

$Percentage = \frac{60^{\circ}}{360^{\circ}}\cdot 100=\frac{1}{6}\cdot 100=.1\bar{6}\cdot 100=16.7\%$

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Example Question #1 : Circles

A sector of a circle contains a center angle that is 36 degrees. What percentage of the circle is the sector?

Possible Answers:

$15\%$

$10\%$

$20\%$

$5\%$

Correct answer:

$10\%$

Explanation:

Proportions can be used to determine the percentage. Let equal the percentage comprised of the sector.

$\frac{36}{360}=\frac{x}{100}$

Cross multiply

3600=360x

Solve for x.

$\frac{3600}{360}=\frac{360x}{360}$

10=x

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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?

Possible Answers:

18π in²

144π in²

4.5π in²

36π in²

9π 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.

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

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

Possible Answers:

$200$

$256\pi$

$100\pi$

$144$

$5\pi$

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 $\frac{1}{4}$ of the circle so it follows:

$Area(DBE)=\frac{400}{4}\pi=100\pi$

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High School Math : Plane Geometry

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #4 : How To Find An Angle In A Polygon

Example Question #5 : How To Find An Angle In A Polygon

Example Question #1 : How To Find The Length Of A Side Of A Polygon

Example Question #1 : Circles

Example Question #1 : How To Find The Angle For A Percentage Of A Circle

Example Question #1 : Circles

Example Question #3 : Circles

Example Question #1 : Circles

Example Question #1 : Plane Geometry

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

All High School Math Resources

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