CK-12 Geometry
Page 38
Review Answers
Approximately times
Approximately
Approximately
Approximately
Approximately
or equivalent
Circles and Sectors
Learning Objectives
Calculate the area of a circle.
Calculate the area of a sector.
Expand understanding of the limit concept.
Introduction
In this lesson we complete our area toolbox with formulas for the areas of circles and sectors. We’ll start with areas of regular polygons, and work our way to the limit, which is the area of a circle. This may sound familiar; it’s exactly the same approach we used to develop the formula for the circumference of a circle.
Area of a Circle
The big idea:
Find the areas of regular polygons with radius .
Let the polygons have more and more sides.
See if a limit shows up in the data.
Use similarity to generalize the results.
The details:
Begin with polygons having and sides, inscribed in a circle with a radius of .
Now imagine that we continued inscribing polygons with more and more sides. It would become nearly impossible to tell the polygon from the circle. The table below shows the results if we did this.
Regular Polygons Inscribed in a Circle with Radius
Number of sides of polygon Area of polygon
As the number of sides of the inscribed regular polygon increases, the area seems to approach a “limit.” This limit is approximately , which is .
Conclusion: The area of a circle with radius is .
Now we extend this idea to other circles. You know that all circles are similar to each other.
Suppose a circle has a radius of .
The scale factor of this circle and the one in the diagram and table above, with radius , is , or just .
You know how a scale factor affects area measures. If the scale factor is , then the area is times as much.
This means that if the area of a circle with radius is , then the area of a circle with radius is .
Area of a Circle Formula
Let be the radius of a circle, and the area.
You probably noticed that the reasoning about area here is very similar to the reasoning in an earlier lesson when we explored the perimeter of polygons and the circumference of circles.
Example 1
A circle is inscribed in a square. Each side of the square is long. What is the area of the circle?
Use . The length of a side of the square is also the diameter of the circle. The radius is .
The area is .
Area of a Sector
The area of a sector is simply an appropriate fractional part of the area of the circle. Suppose a sector of a circle with radius and circumference has an arc with a degree measure of and an arc length of .
The sector is of the circle.
The sector is also of the circle.
To find the area of the sector, just find one of these fractional parts of the area of the circle. We know that the area of the circle is . Let be the area of the sector.
Also,
Area of a Sector
A circle has radius . A sector of the circle has an arc with degree measure and arc length .
The area of the sector is .
Example 2
Mark drew a sheet metal pattern made up of a circle with a sector cut out. The pattern is made from an arc of a circle and two perpendicular radii.
How much sheet metal does Mark need for the pattern?
The measure of the arc of the piece is , which is of the circle.
The area of the sector (pattern) is .
Lesson Summary
We used the idea of a limit again in this lesson. That enabled us to find the area of a circle by studying polygons with more and more sides. Our approach was very similar to the one used earlier for the circumference of a circle. Once the area formula was developed, the area of a sector was a simple matter of taking the proper fractional part of the whole circle.
Summary of Formulas:
Area Formula
Let be the radius of a circle, and the area.
Area of a Sector
A circle has radius . A sector of the circle has an arc with degree measure and arc length .
The area of the sector is square units.
Points to Consider
When we talk about a limit, for example finding the limit of the areas of regular polygons, how many sides do we mean when we talk about “more and more?” As the polygons have more and more sides, what happens to the length of each side? Is a circle a polygon with an infinite number of sides? And is each “side” of a circle infinitely small? Now that’s small!
In the next lesson you’ll see where the formula comes from that gives us the areas of regular polygons. This is the formula that was used to produce the table of areas in this lesson.
Review Questions
Complete the table of radii and areas of circles. Express your answers in terms of .
Radius (units) Area (square units)
1a. ?
1b. ?
1c. ?
1d. ?
Prove: The area of a circle with diameter is
A circle is inscribed in a square.
The yellow shaded area is what percent of the square?
The circumference of a circle is . What is the area of the circle?
A center pivot irrigation system has a boom that is long. The boom is anchored at the center pivot. It revolves around the center pivot point once every three days, irrigating the ground as it turns. How many hectares of land are irrigated each day?
Vicki is cutting out a gasket in her machine shop. She made a large circle of gasket material, then cut out and removed the two small circles. The centers of the small circles are on a diameter of the large circle. Each square of the grid is .
How much gasket material will she use for the gasket?
A security system scans all points up to from is base. It scans back and forth through an angle of .
How much space does the system cover?
A simplified version of the international radiation symbol is shown below.
The symbol is made from two circles and three equally spaced diameters of the large circle. The diameter of the large circle is , and the diameter of the small circle is . What is the total area of the symbol?
Chad has of fencing. He will use it all. Which would enclose the most space, a square fence or a circular fence? Explain your answer.
Review Answers
1a. 1b. 1c. 1d.
Approximately
Approximately
Approximately
Approximately
Approximately
The circular fence has a greater area. Square:
Circle:
Regular Polygons
Learning Objectives
Recognize and use the terms involved in developing formulas for regular polygons.
Calculate the area and perimeter of a regular polygon.
Relate area and perimeter formulas for regular polygons to the limit process in prior lessons.
Introduction
You’ve probably been asking yourself, “Where did the areas and perimeters of regular polygons in earlier lessons come from?” Or maybe not! You might be confident that the information presented then was accurate. In either case, in this lesson we’ll fill in the missing link. We’ll derive formulas for the perimeter and area of any regular polygon.
You already know how to find areas and perimeters of some figures—triangles, rectangles, etc. Not surprisingly, the new formulas in this lesson will build on those basic figures—in particular, the triangle. Note too that we will find an outstanding application of trigonometric functions in this lesson.
Parts and Terms for Regular Polygons
Let’s start with some background on regular polygons.
Here is a general regular polygon with sides; some of its sides are shown.
In the diagram, here is what each variable represents.
is the length of each side of the polygon.
is the length of a “radius” of the polygon, which is a segment from a vertex of the polygon to the center.
is the length of one-half of a side of the polygon
is the length of a segment called the apothem—a segment from the center to a side of the polygon, perpendicular to the side. (Notice that is the altitude of each of the triangles formed by two radii and a side.)
The angle between two consecutive radii measures because congruent central angles are formed by the radii from the center to each of the vertices of the polygon. An apothem divides each of these central angles into two congruent halves; each of these half angles measures .
Using Trigonometry with the Regular Polygon
Recall that in a right triangle:
In the diagram above, for the half angles mentioned,
is the length of the opposite side
is the length of the adjacent side
is the length of the hypotenuse
Now we can put these facts together:
Perimeter of a Regular Polygon
We continue with the regular polygon diagrammed above. Let be the perimeter. In simplest terms,
Here is an alternate version of the perimeter formula.
Perimeter of a regular polygon with sides and a radius long:
One more version of the perimeter formula applies when the polygon is inscribed in a “unit circle,” which is a circle with a radius of .
Perimeter of a regular polygon with sides inscribed in a unit circle:
Example 1
A square has a radius of . What is the perimeter of the square?
Use , with and .
Notice that a side and two radii make a right triangle.
The legs are inches long, and the hypotenuse, which is a side of the square, is inches long.
Use
The purpose of this example is not to calculate the perimeter, but to verify that the formulas developed above “work.”
Area of a Regular Polygon
The next logical step is to complete our study of regular polygons by developing area formulas.
Take another look at the regular polygon figure above. Here’s how we can find its area, .
Two radii and a side make a triangle with base and altitude .
There are of these triangles.
The area of each triangle is .
The entire area is .
Area of a regular polygon with apothem :
We can use trigonometric functions to produce a different version of the area formula.
(remember that )
(remember that and )
Area of a regular polygon with sides and radius :
One more version of the area formula applies when the polygon is inscribed in a unit circle.
(remember that )
Area of a regular polygon with sides inscribed in a unit circle:
Example 2
A square is inscribed in a unit circle. What is the area of the square?
Use with .
The square is a rhombus with diagonals long. Use the area formula for a rhombus.
Comments: As in example 1, the purpose of this example is to show that the new area formulas do work. We can confirm that the area formula gives a correct answer because we have another way to confirm that the area is correct.
Lesson Summary
The lesson can be summarized with a review of the formulas we derived.
Perimeter Area
Any regular polygon
Any regular polygon
Regular polygon inscribed in a unit circle
Points to Consider
We used the concept of a limit in an earlier lesson. In the Lesson Exercises, you will have an opportunity to use the formulas from this lesson to “confirm” the circumference and area formulas for a circle, which is the “ultimate” regular polygon (with many, many sides that are very short).
Review Questions
Each side of a regular hexagon is long.
What is the radius of the hexagon?
What is the perimeter of the hexagon?
What is the area of the hexagon?
A regular and a regular are inscribed in a circle with a radius of .
Which polygon has the greater perimeter?
How much greater is the perimeter?
Which polygon has the greater area?
How much greater is the area?
A regular is inscribed in a unit circle. The area of the , rounded to the nearest hundredth, is . What is the smallest possible value of ?
Review Answers
The
The
Geometric Probability
Learning Objectives
Identify favorable outcomes and total outcomes.
Express geometric situations in probability terms.
Interpret probabilities in terms of lengths and areas.
Introduction
You’ve probably studied probability before now (pun intended). We’ll start this lesson by reviewing the basic concepts of probability.
Once we’ve reviewed the basic ideas of probability, we’ll extend them to situations that are represented in geometric settings. We focus on probabilities that can be calculated based on lengths and areas. The formulas you learned in earlier lessons will be very useful in figuring these geometric probabilities.
Basic Probability
Probability is a way to assign specific numbers to how likely, or unlikely, an event is. We need to know two things:
the total number of possible outcomes for an event. Let’s call this .
the number of “favorable” outcomes for the event. Let’s call this .
The probability of the event, call it , is the ratio of the number of favorable outcomes to the total number of outcomes.
Definition of Probability
Example 1
Nabeel’s company has holidays each year. Holidays are always on weekdays (not weekends). This year there are weekdays. What is the probability that any weekday is a holiday?
There are weekdays in all.
of the weekdays are holidays
Comments: Probabilities are often expressed as fractions, decimals, and percents. Nabeel can say that there is a chance of any weekday being a holiday. Note that this is (unfortunately?) a relatively low probability.
Example 2
Charmane has four coins in a jar: two nickels, a dime, and a quarter. She mixes them well. Charmane takes out two of the coins without looking. What is the probability that the coins she takes have a total value of more than ?
in this problem is the total number of two-coin combinations. We can just list them all. To make it easy to keep track, use these codes: (one of the nickels), (the other nickel), (the dime), and (the quarter).
Two-coin combinations:
There are six two-coin combinations.
Of the six two-coin combinations, three have a total value of more than . They are:
The probability that the two coins will have a total value of more than is .
The probability is usually written as , or . Sometimes this is expressed as “a chance” because the probability of success and of failure are both .
Geometric Probability
The values of and that determine a probability can be lengths and areas.
Example 3
Sean needs to drill a hole in a wall that is wide and high. There is a by rectangular mirror on the other side of the wall so that Sean can’t see the mirror. If Sean drills at a random location on the wall, what is the probability that he will hit the mirror?
The area of the wall is . This is .
The area of the mirror is . This is .
The probability is .
Example 4
Ella repairs an electric power line that runs from Acton to Dayton through Bar
ton and Canton. The distances in miles between these towns are as follows.
If a break in the power line happens, what is the probability that the break is between Barton and Dayton?
Approximately .
Lesson Summary
Probability is a way to measure how likely or unlikely an event is. In this section we saw how to use lengths and areas as models for probability questions. The basic probability ideas are the same as in non-geometry applications, with probability defined as:
Points to Consider
Some events are more likely, and some are less likely. No event has a negative probability! Can you think of an event with an extremely low, or an extremely high, probability? What are the ultimate extremes—the greatest and the least values possible for a probability? In ordinary language these are called “impossible” (least possible probability) and “certain” or a “sure thing” (greatest possible probability).