# Circles

This page is devoted to investigating the properties and features of circles.

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

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

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.

*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.

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

Why is the area measurement found in the Entity Information window off by almost 1 meter?

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)

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.

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