CIRCLES 1: FIRST PRINCIPLES

Intro

When I wanted to make a circular flower bed around the birch tree in my front yard, I tied a rope to the handle of my spade, then tied the other end of the rope around the tree, with exactly the right length of rope between the spade handle and the tree to make the right size flower bed. I then walked around the tree planting the spade into the earth to make the outline of the flower bed's circular edge or circumference.

I did it this way because I knew the definition of a circle. Before we look at the definition though, let's discuss the word equidistant. We can see that the first part of it looks a lot like the word equal -- and that's exactly what it means. So equidistant means " equally distant". Now here's the definition.

 A circle is a path or locus of all points in the plane that are equidistant from a center point.The radius of the circle is the distance from the center to any point on the circumference.

In my project, the tree was the center of the circle and the length of rope between the tree and the spade handle was the radius of my circular flower bed.

compasses

When we make a circle using compasses, first, we set the radius and then, we choose the center point. Then we dig the point of the compasses into the paper -- and sometimes the desktop -- and turn the penciled arm around until we've plotted every point on the page that is exactly radius distance away from the center.

Hint: Notice in the picture how the pencil is inserted so as to make the arms of the compasses equal in length. If you have trouble making circles -- if your compasses keep slipping out of control -- change the height of the pencil to equalize the arms and make sure the center point is well planted in the paper (or desktop -- but not your hand!!).

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Circle or Disc?

Note: When we call something a circle, we're really describing the circumference, the edge or perimeter of a disc. So, when discussing area, segments and sectors we should properly refer to a disc rather than a circle. Think of your compact discs. You'd never call them circles -- would you? A circle forms the edge of a disc. However, since we usually refer to both the circumference and the area enclosed within it as a circle, we can use either term.

Circle Definitions

Chord: The line segment joining two points on the circumference of a circle.
Diameter: The longest chord. It passes through the center and equals 2 times the radius.
Segment: The two parts or surfaces into which a chord divides a disc.
Arc: The curved part of the circumference cut off by the end points of a chord.
Sector: A sector is the area of a disc bounded by two radii and the arc between them.
Note: We name circle segments with three letters: one at each end of the chord and another on the circumference between the initial two (as shown in the diagram).
An arc is named with three points too, like a segment. Each chord creates two arcs: a major arc -- the larger of the two, and a minor arc -- the smaller of the two arcs defined by the chord.
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Circumference of a Circle

Discovery Activity

On a piece of cardboard, using your compasses, draw a circle with a radius of 1½ inches.
Carefully draw a diameter to pass exactly through the center hole.
Now, carefully cut out the circle and mark a point P on the circumference.
Next, measure the circumference by rolling your circle along the edge of your ruler
as shown -- until point P returns to its original position.
Now divide the circumference measure by the diameter (3 inches) and record the results.
(round to 2 decimal places.)

Did you find the circumference measures about 9.5 inches? If so, when you divide by the 3 inch diameter, you will get something close to 3.17.
If we repeat this activity with zillions of circles with different diameters, the quotient C/d will always equal 3 and a small decimal.

This number is called pi (pronounced pie). It is a Greek letter (the equivalent of p in the English alphabet) and here is the symbol we use to represent it.

Pi is an irrational number -- which means that the decimal part of the number never ends and never repeats. Taken to 5 decimal places, pi = 3.14159. The ancient monument building Egyptians often used or 3.14286 -- in our work with circles, we usually set pi = 3.14.

So here's what we've discovered:

The circumference of a circle is equal to pi times the diameter
or pi times twice the radius.

If we're given the measure of the Circumference, we can find the lengths of the diameter and radius by division. We know that .

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Example: Find the Circumference for a circle with diameter = 7 inches.
Solution: Since , C = 3.14 × 7 = 21.98 inches.

Example: Find the diameter and radius of a circle with Circumference = 47.1 inches.

Solution: Since , d = 47.1 ÷ 3.14 = 15 inches. Radius = ½ (d) = 7.5 inches.

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Area of a Circle

In order to show that the area of a cicle is , let's turn it into a parallelogram
with base = half the circumference and height or altitude = radius. Here's how we do it.

The area of a circle or disc is pi × radius²

Remember to use SQUARE UNITS for area.

If we divide the Area of a circle or disc by pi, we get r².
So to find the radius, we take the square root of this answer.

Example: Find the area for a circle with diameter = 7 inches.

Solution: Since radius = 3.5 inches, , A = 3.14 × (3.5)² = 38.47 square inches.

Example: Find the diameter and radius of a circle with Area = 50.24 feet.

Solution: Since r² = 50.24 ÷ 3.14 = 16 inches.
We know that 4² = 16, so the radius = 4 inches, the diameter = 8 inches.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again.

Practice

Round answers to 2 decimal places.

1) Use the diagram to match the list on the right with that on the left:

 BZC a) a radius XY b) a sector of a circle AOX c) a major arc or segment of a circle XDY d) a minor arc or segment of a circle OA e) a semicircle BDC f) a diameter

2) Calculate the Circumference and Area for these circles.
Set pi = 3.14. Round answers to 2 decimal places.

 a) diameter = 5.7 feet b) radius = 7.3 inches c) ¼ C = 4.71 inches d) ½ C = 18.84 yards

3) Calculate the perimeter and area of these figures. Round answers to 2 decimal places.
reminder: perimeter is distance around the outside of the figure.

4) Calculate the area of the blue region. Round answers to 2 decimal places. (solution)

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Solutions

1) Use the diagram to match the list on the right with that on the left:
 BZC d) a minor arc or segment of a circle XY f) a diameter AOX b) a sector of a circle XDY e) a semicircle OA a) a radius BDC c) a major arc or segment of a circle

2) Calculate the Circumference and Area for these circles.
Set pi = 3.14. Round answers to 2 decimal places.

 a) diameter = 5.7 feet so radius = 2.55 feetC = 5.7 × 3.14 = 17.90 feet A = 3.14 × (2.55)² = 20.42 square feet b) radius = 7.3 inchesC = 2 × 7.3 × 3.14 = 45.84 inches A = 3.14 × (7.3)² = 167.33 square inches c) ¼ C = 4.71 inches,C = 4 × 4.71 = 18.84 A = 3.14 × (3)² = 28.26 square inches d) ½ C = 18.84 yardsC = 2 × 18.84 = 37.68 A = 3.14 × (6)² = 113.04 square yards

3) Calculate the perimeter and area of these figures. Round answers to 2 decimal places.

 a) P = ½ C + d P = ½ (3.14× 4.2) + 4.2 P = 10.79 inchesd = 4.2, so r = 2.1A = ½ (3.14)(2.1)² A = 6.92 square inches. b) P = ¼ C + 2r P = ¼ (3.14 × 2.86) + 2.86 P = 5.11 inchesA = ¼ (3.14)(1.43)² A = 1.61 square inches. c) P = ¾ C + 2r P = ¾ (3.14 × 0.5) + 0.5 P = 1.68 feetA = ¾ (3.14)(0.25)² A = 0.15 square feet.

4) Calculate the area of the blue region. Round answers to 2 decimal places.

The area of the blue region is the area of the rectangle less the area of the circle.
Since the diameter of the circle = 6 inches, the radius = 3 inches.
Rectangle Area = l × w = 6 × 15 = 90 in²
Circle Area = = 3.14 × 3² = 28.26 in²

The area of the blue region is 90 - 28.26 = 61.74 in².

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