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Area & Perimeter
Circles - A&P
Fun with Circles
Quads - A&P
Decimals to Inches
Sphere & Torus
Sphere Vol Formula
Cylinders Vol & SA
Net 3-4-5 Prism
Net Right Tri Prism
Net Hexagon Prism
Net Octagonal Prism
Nets for 3 Pyramids
Lateral SA and Vol
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This page is devoted to investigating the properties and features of circles.
You should be able to
this page and explain what you
to your teacher.
Vocabulary Development (hands-on and technology supported)
Circle, chord, radius, diameter,circumference, area, sector, segment, secant line, tangent line, inscribed, circumscribed
Understand the features and properties of a circle
SketchUp, calculator, pencil, square sticky notes, string
You will study circles through activities with and with out technology.
You will find lengths, measures, and areas.
A circle is named after its center point (upper case).
While polygons have sides, circles have arcs.
A piece of a circle is called an arc.
A minor arc is less than half of the circle.
We use the term “circumference” for the length of a curved segment.
Choose this Template Window | Model Info - set the precision as shown here.
What is a circle?
Note that a circle is an infinite set of points IN A PLANE that are equidistant from a given point.
The radius is the shortest segment from the center point to the circle.
A chord is a segment that connects two distinct points of the circle.
A diameter is a chord that includes the center point.
Half of a circle is called a semi-circle.
The example shown here has a radius of 3 meters.
Sectors and Segments
The segment’s boundaries are an arc and a chord that share the arc’s endpoints.
The sector’s boundaries are the arc and the two radii that share the arc’s endpoints.
PART I - CIRCUMFERENCE AND ARC LENGTH
- Estimate the Circumference for a circle with a radius of 5 meters.
Use what you know about length to find the circumference “perimeter” of the circle.
Use a piece of string that is equal to either the diameter or the radius.
Use a “piece sign” with three sectors as a MEMORY TRIGGER.
*It is not possible to find the exact circumference with SketchUp because a circle is actually a 24-gon (you can change the number of sides, but it will still be a polygon).
- Calculate the circumference to two decimal places
C ≈ 3.14 * diameter
C ≈ 3.14 * 10 m = 31.4 m Exact answer is 10π meters
- ARC MEASURE (degrees)
Remember that arcs are measured with degrees. One complete arc measures 360 degrees.
Try to figure out how arc measure is related to an arc of a given fraction of a circle by completing the “arc measure” in the chart below.
- ARC LENGTH ( linear measure)
Remember that length is measured with a linear unit like inches, feet, or meters.Try to figure out how to use what you know about circumference and arc measure to find a given “arc length” in the chart below.
- Find a Pattern from Examples
Fraction of a circle
Sketch of arc
Arc length (2 decimals)
Half a circle with radius 4 meters
½ of 360°
Length = ½*C
(1/2)(3.14) (8m) = 12.56 m
A third of a circle with radius 11 inches
1/3 of 360°
Length = 1/3*C
(1/3)(3.14) (22 “) ≈ 22.03”
An eighth of a circle with radius 4 feet
1/8 of 360°
Length = 1/8*C
(1/8)(3.14)(8’) = 3.14’
A fraction (1/n) of a circle
1/n of 360°
Length = 1/n*C
(1/n)(3.14)(d) linear units
+Note that the arc tool in SketchUp uses three distinct points to create an arc.
Part II – AREA
- Estimate the Area
Hint: Use what you know about the area of a square to estimate how many green “radius squares” will cover the entire circle.
Construct a circle with a compass using a radius equal to the side length of a square sticky note.
Cut up one sticky note at a time to “cover” the circle.
How many sticky notes do you need?
- Calculate the Area of a circle with a radius of 5 cm (to two decimal places).
A ≈ 3.14 x r x r
A ≈ 3.14 * 5 m* 5 m
A ≈ 78.5 m2
Exact answer is 25π m2
Why is the area measurement found in the Entity Information window off by almost 1 meter?
What if we only want the area a sector of the circle?
1. Find the area of a semi-circle with radius 24 inches.
A ≈ 3.14(24in)(24in)(.5) = 904 square inches
2. Find the area of a quarter of a circle whose radius is 13 inches.
A ≈ 3.14(13in)(13in)(.25) = 132.665 square inches
PART III – SPECIAL LINES and FIGURES
Secant and Tangent Lines (shown here as segments)
When lines intersect circles in a given plane, they can do so in only one of two ways.
A line that passes through a circle at two
points is called a secant line. A secant line contains a chord.
A tangent line intersects the circle at one point called the point of tangency. Notice that at the point of tangency, the radius and tangent line are perpendicular.
Construction Tip - Be sure to use the perpendicular guide (purple line) as you draw the tangent so it is perpendicular to the radius. DO NOT EYEBALL IT.
Inscribed Polygons // Circumscribed Circles - A polygon is inscribed in a circle if all vertices of the polygon lie on the circle and all sides of the polygon lie INSIDE the circle.
A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle and all sides of the polygon lie OUTSIDE the circle.
Page Updated - June 2014
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