Analytical Geometry Circles and Volume Circles and Volume There is something so special about a circle. It is a very efficient shape. There is no beginning, no end. Every point on the edge is the same distance from the center. No one is left out, everyone is the same. Without circles, we would all be grounded. No place to go, no world to explore. Few games to play, few planets to gaze at. How are circles used in your life? Do you ride a car or bus to school or work? Do you play any sports? Do you study stars and far off planets? Do you work or live near a farm? These are just a few examples of where circles and spheres are used in our everyday lives. They are all around us. What would we do without them? Essential Questions What are the different theorems that apply to circles? How can I use the circle theorems to solve problems? How do I find arc lengths of a circle? How do I find the area of a sector of a circle? How do I find the volume of a sphere? How can I use volume to find other dimensions of a sphere? Module Minute Key Words In this module we will use circle theorems to prove that all circles are similar. Chords, radii's, diameters, and angles within and around the circle all relate to each other. We will identify and describe how inscribed angles (which are angles inside of circles) relate to the radius of a circle as well as the chords that make up the angles. Central angles and circumscribed angles also have relationships with the diameter and right triangles. The radius and circumference of a circle can be used in a proportion with a central angle to find the length of an arc. The area of a circle can help us find the area of a sector and the radius will come in handy when finding the volume of a sphere. Central Angle An angle whose vertex is at the center of a circle. Inscribed Angle An angle whose vertex is on the circle and whose sides contain chords of a circle. Arc An unbroken part of a circle. Chord A segment whose endpoints are on a circle. Tangent Line A line in the plane of a circle that intersects a circle at only one point. Secant Line A line in the plane of a circle that intersects a circle at exactly two points. Secant Segment A segment that contains a chord of a circle and has exactly one endpoint outside of the circle. Inscribed Polygon A polygon whose vertices all lie on a circle. Inscribed Circle A circle enclosed in a polygon, where every side of the polygon is a tangent to the circle. Circumscribed Circle A circle containing an inscribed polygon. A handout of these key words and definitions is also available in the sidebar. What To Expect Circles Assignment Arcs Quiz Inscribed Quadrilateral Discussion Angles and Segments Quiz Arcs Investigation Task Circles and Volume Test Circles and Volume Project To view the standards from this unit, please download the handout from the sidebar. Circle Theorems
When putting lines and circles together, there are many relationships and theorems that result. The first thing we need to do is recall the names of different lines as they are placed in and around a circle. Watch the video below to learn about the Language and Notation of the Circle. Scroll through the album below to learn more about the language of Circles. Tangent Circles Tangent circles are circles that intersect at one point only. A tangent line that the two circles share is called a common tangent. Circles can have common internal tangents and common external tangents. Internal Tangent External Tangent Tangent lines that are on the outside of two circles are external and tangent lines that come between the two circles are called internal. Example 1 Give the name of the segment or line that represents each of the following terms. Notice that some of the lines have more than one name. For example, the diameter is also a chord because it passes through the circle and its endpoints are on the edge of the circle. Also, there can be more than one of each type of line. In fact there can be an infinite number! Think of how many radii's (plural for radius) that you could draw in a circle. Example 2 Now let's try going the other way. Identify each of the following segments as tangent, chord, radius, diameter, or secant. SPECIAL NOTE: In the Example 2 Video below, the labels in the image are misplaced. The 24 and 25 need to be switched on the image. Side AC should be 25 and side BC should be 24. There are many theorems that relate lines and line segments to circles. Be sure that you are familiar with all of the parts of the circles in the above two examples before moving on. Similar Circles Any shapes that can undergo a series of transformations and end up on top of an identical shape are said to be similar. Do you think that circles are similar to each other? Circle Theorems Now, let's take a look at the first few circle theorems. Perpendicular Tangent Theorem If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. In example 1 below, since line DM is tangent to circle C at point D, then DM is CD.
We can also switch this theorem around and write what is called the converse. Converse of Perpendicular Tangent Theorem If a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle. If CD DM at point D, then line DM is tangent to circle C. Knowing this theorem helps us understand the relationship between a radius and a tangent line. This information can then be used to solve problems about a circle. Let's look at the examples below. Now we can see why we can't always trust what we see in a picture! Constructing a Tangent to a Circle Constructing a tangent line is very similar to constructing perpendicular lines since a tangent is perpendicular to the radius at the point of tangency. To create your own tangent line, follow the steps below: Circles Assignment Select the "Circles Assignment" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment when finished. Arcs The distance around the outside of a circle, the circumference, can be broken up into many pieces. Each of these pieces is called an arc and is labeled based on its size measured in degrees. (Remember that a circle is 360.)
A minor arc is an arc that is less than 180. We use two end points to name it: A major arc is an arc that is greater than 180. We use three letters to name it: A semicircle is an arc that is equal to 180. We use three letters to name it: (Spoken: arc AB.) Measuring Arcs The measure of a central angle is equal to the minor arc. Since <GFE = 95, then = 95. How Do You Find the Measure of an Arc? The measure of a major arc is 360 minus the associated minor arc. For example: Example 1: = 360 95 = 265 Tell if each arc is major, minor, or a semicircle. Find the measure of the arc.
Try to answer these on your own and then roll over the ANSWERS below to understand the measures of this arc. ANSWER ANSWER ANSWER ANSWER Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measure of the two arcs. Example 2: Find the measure of the given arcs.
= 60 + 82 = 142 = 142 + 100 = 242 = 360 242 = 118 Congruent Arcs Arcs are considered to be congruent if they have the same measure and the same radius. Example 3: Find the measure of and. Are the arcs congruent? Measures of and? ANSWER The arcs since they have the same measure and they are in the same circle. Example 4: Find the measure of and. Are the arcs congruent?
Measures of and? ANSWER The arcs since they have the same measure and their radii's of the circles are congruent. Example 5: Find the measure of and. Are the arcs congruent? Measures of and? ANSWER The arcs since they do not have congruent radii's. Chords Congruent Chord Theorem: In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. Example 1: Find the m.
ANSWER Perpendicular Bisector Theorem: If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Example 2: Tell whether is a diameter of the circle. Yes, because is a perpendicular bisector of. Example 3: Tell whether is a diameter of the circle. ANSWER Perpendicular Diameter Theorem: If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.
Example 4: Find the measure of. Since, is the diameter, it bisects. If is 5 then is also 5 making 10. Equidistant Chord Theorem: If the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. Example 5: Find the measure of.
ANSWER Inscribed Angles and Polygons Measure of an Inscribed Angle Theorem: The measure of an inscribed angle is one half the measure of its intercepted arc. Example 1: What is the measure of? ANSWER Example 2: What is the measure of <S in the image above in example 1? ANSWER Congruent Inscribed Angles Theorem: If two inscribed angles intercept the same arc, then the angles are congruent.
Example 3: Find the measure of <RTS. ANSWER Example 4: In example 3, m<qrt is 48, what is the m<qst? ANSWER Inscribed Right Triangle Theorem: A right triangle is inscribed in a circle if and only if the hypotenuse of the triangle is a diameter of the circle.
This makes sense because of our inscribed angle theorem. If the inscribed angle is half of the intercepted arc, and in this case, the intercepted arc is a semi circle, then the angle must be 90 degrees. Example 5: Find the measure of <NOP. because it's endpoints are the of circle M. Inscribed Quadrilateral Theorem: A quadrilateral can be inscribed in a circle if its opposite angles are supplementary.
Watch the video below to learn more about the theorems. Self Check: In this circle, What is the measure of angle QPR? 18, 24, 36, or 72 SOLUTION Arcs Quiz It is now time to complete the "Arcs" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly. Inscribed Quadrilateral Discussion Use the theorems that you that you have learned about inscribed angles and intercepted arcs to explain why inscribed quadrilaterals have opposite angles that are supplementary. A rubric for the discussion is located in the sidebar. Angles and Segments in a Circle
Angles and Segments in a Circle Tangent Chord Theorem: If a tangent and a chord intersect, the measure of each angle formed is half the measure of its intercepted arc. In the diagram, is an example of a chord and is a tangent. is a major arc. The chord and the tangent line intersect on the circle. m<1 is 70 and m<2 is 110. Angles Inside the Circle Theorem: If two chords intersect, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. In the diagram, and are both chords. Angles Outside the Circle Theorem: If two secants or tangents intersect to form an angle outside of a circle, the angle is equal to half of the difference of the intercepted arcs.
In this example, the circle is intersected by two secants that form an angle outside of the circle. Line Segments in a Circle Segments of Chords Theorem: If two chords intersect in the interior of a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. Example 1: Find x.
8(6) = x (12) 48 = 12x 4 = x Segments of Secants Theorem: If two secants segments share the same external endpoint, then the product of one secant segment and its outer piece is equal to the product of the other secant and its outer piece. Segments of Secants and Tangents Theorem: If a secant and a tangent intersect outside a circle, the product of the outside segment of the secant and the whole secant is equal to the tangent segment squared.
Example 2: Find the value of x. 10 2 = 5(5+x) 100 = 25 + 5x 75 = 5x? = x Example 3: Find the value of x.
36 2 = 24(24 + x) 1296 = 576 + 24x 720 = 24x? = x Arc Length and Sectors Investigation Assignment: Arcs Complete the Arcs Task found in the sidebar. This is an introductory task that will help you discover some of the learning objectives of this module. Submit your answers when you are finished. We have talked a lot about the measurement of arcs but how can we actually find how long an arc is? When you have a piece of pie, how do you know if your piece is bigger than the others? We can use proportions of a circle's circumference and area to find the length of an arc and the area of a sector (your piece of pie.) Let's review a few key facts. The circumference of a circle is the distance around the outside. The formula is. The length of an arc is the portion of a circle's circumference. Think of this as the length of the crust on your piece of pie. A sector of a circle is the region bounded by two radii of the circle and their intercepted arc. In other words, your slice of pie. To find the sector, we need to recall that area of a circle is. Let's review how to find circumference and area of a circle. Watch the video below. Example 1: Find the circumference of a circle with radius 13 ft.
Example 2: Find the area of a circle with diameter 12 ft. Recall that the radius is half of the diameter. Arc Length Recall that we compared arc length to the crust of our slice of pie. When you take a slice of pie, you are taking a portion of the whole pie. If your slice of pie is ¼ of the whole pie, then it makes sense that your crust will be ¼ of the circumference of the pie. We can use this logic to set up a proportion. Example 3: Find the length of. Now Solve:
Example 4: Find the length of. Now Solve: Area of a Sector: Example 5: Find the area of the sector defined by.
Now Solve: (Area of Sector) (9) = (100 ) (4) (Area of Sector) (9) = 400 Area of Sector =???? Volume Have you ever wondered how much space is inside of a basketball? What about how much ice cream can be packed into an ice cream cone? Using simple mathematical formulas, we can answer these questions as well as many others dealing with the volume of different solid shapes. Watch the video below to review solid geometry before moving on to the examples. Volume of a Sphere To find the volume, which is the space inside, of a sphere, we only need to know one measurement. The measurement that we need is the radius. With that one single measurement we can find out the cubic space that would fill any size sphere.
Example 1: Let's find the volume of a sphere that has a radius of 9. Volume of Cones and Cylinders If you had a cone and a cylinder that had the same radius, how many times could you pour water from the cone into the cylinder before the cylinder would be full? To answer this we need to know the volume of each the cone and cylinder. Volume of a Cylinder Volume of a Cone V = Area of the Base x Height V = volume of a cylinder V = B x h V = V = (B x h of cylinder) V = Volume of Rectangular Prisms and Pyramids What about the same question using a rectangular prism and a pyramid. Do you think they will have a similar relationship? Volume of Rectangular Prism Volume of Right Pyramid
V = Area of Base x height V = B x h V = volume of prism V = lwh V = lwh Notice the formulas have the same relationship! The volume of a right pyramid is the volume of a rectangular prism. Cross Sections of 3D Objects Watch the following videos to see the shapes that result when we slice through solid objects by clicking on each title. Vertical Slice of a Rectangular Pyramid video Ways to Cut a Cube video Rotating 2D Shapes in 3D video Angles and Segments Quiz It is now time to complete the "Angles and Segments" quiz. You will have a limited amount of time to complete your quiz; please plan accordingly Module Wrap Up In this module you were responsible for completing the following assignments. Circles Assignment Arcs Quiz
Review Inscribed Quadrilateral Discussion Angles and Segments Quiz Arcs Investigation Task Circles and Volume Test Circles and Volume Project Now that you have completed the initial assessments for this module, review the lesson material with the practice activities and extra resources. Then, continue to the next page for your final assessment instructions. Standardized Test Preparation The following problems will allow you to apply what you have learned in this module to how you may see questions asked on a standardized test. Please follow the directions closely. Remember that you may have to use prior knowledge from previous units in order to answer the question correctly. If you have any questions or concerns, please contact your instructor. Final Assessments Circles and Volume Final Module Test It is now time to complete the "Circles and Volume" Test. Once you have completed all self assessments, assignments, and the review items and feel confident in your understanding of this material, you may begin. You will have a limited amount of time to complete your test and once you begin, you will not be allowed to restart your test. Please plan accordingly. Circles and Volume Project Select the "Circles and Volume Project" Handout from the sidebar. Record your answers in a separate document. Submit your completed assignment when you are finished.