Circles

This page is devoted to investigating the properties and features of circles.
You should be able to read this page and explain what you read to your teacher.

Vocabulary Development (hands-on and technology supported)
Circle, chord, radius, diameter,circumference, area, sector, segment, secant line, tangent line, inscribed, circumscribed

Learning Objective: Understand the features and properties of a circle
Materials: SketchUp, calculator, pencil, square sticky notes, string


SPECIFIC TASKS
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.

Tangram Window Info.jpg

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.
sketchup wiki Parts of Circle.jpg
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.
Skechup Segment Sector.jpg


PART I - CIRCUMFERENCE AND ARC LENGTH


Task 1 - 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.
Sketchup Circle Peace Sign.jpg
*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).

Task 2 - Calculate the circumference to two decimal places
C ≈ 3.14 * diameter
C ≈ 3.14 * 10 m = 31.4 m Exact answer is 10π meters

Task 3 - 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.

Task 4 - 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.

Task 5 - Find a Pattern from Examples
Fraction of a circle
Sketch of arc
Arc measure
Arc length (2 decimals)
Half a circle with radius 4 meters

½ of 360°
180°
Length = ½*C
(1/2)(3.14) (8m) = 12.56 m
A third of a circle with radius 11 inches

1/3 of 360°
120°
Length = 1/3*C
(1/3)(3.14) (22 “) ≈ 22.03”
An eighth of a circle with radius 4 feet

1/8 of 360°
45°
Length = 1/8*C
(1/8)(3.14)(8’) = 3.14’
A fraction (1/n) of a circle
1/n of 360°
or
360°/n
Length = 1/n*C
or
(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


Task 1 - 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.
Sketchup Radius Square.jpg
Hands-on Activity
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?

Task 2 - 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?
sketchup Circle Area.jpg
Area Challenge
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)
Sketchup Secant Tangent Lines.jpg
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 distinct 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.

Additional Information

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.
Sketchup Inscribed Polygons.jpg
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.
Sketchup Circle in Pentagon.jpg
Page Updated - June 2014