Subject: Geometry

Circles, More Than Inventing The Wheel

A Perfect Circle

A circle, according to Schaum’s Outlines Geometry, is the set of all points in a plane that are the same distance from the centre. Here is a figure showing the circle and some of its friends:


Figure 1
Figure 1: A Circle, a closed Squiggle, a Cylinder



Circumference of a Circle

The circumference is the smallest possible immediate distance around a circle. I remember this definition by envisioning a Circus Tent, in birds' eye view. The word circumference almost sounds like "circus" and "fence", so I think of the fence that goes all the way around the circus tent, keeping people out. Only this would be a perfect circus fence, with all the posts equi-distant from the center pole holding up the tent. I'm visual, if you're not, ignore the creepy circus reference.

Figure 2
Figure 2: Circumference Shown Compared to Not Circumference

Terms to Commit to Memory

Here are a few terms and definitions regarding circles. Figure 3 will show you how they relate to each other and their parent circle. By working through the examples and testing yourself you will learn the definitions and be able to recall them from memory.


Figure 3
Figure 3: Important Terms to Remember for the BIG Test


Circumference: The smallest possible distance around a circle.

Chord: A straight line joining two points on the circumference of a circle.

Secant: A straight line joining any two points on a circle and continuing beyond those points in both directions.

Tangent: A straight line that just touches the edge of the circle at one point only. Tangents are perpendicular to radii.

Arc: An arc is a portion of the circumference of a circle between and including two points. Arcs can be minor or major, lesser or greater in length than half the circumference.

Diameter: A chord that passes through the middle of the circle.

Radius: The straight length or distance between the centre of a circle and a point on its circumference. The radius of a circle is exactly half of that circle's diameter. The plural of radius is radii (pronounced ‘ray-dee-eye’). It follows as a logical axiom (an axiom is a fundamental law or truth) that all radii of a given circle are congruent.

Semicircle: An arc measuring exactly half the circumference of a circle. Literally half a circle.

Central Angle: An angle formed by two radii of the same circle (not two overlapping circles).

Sector: An enclosed portion of the circle made up of two radii and the part of the circumference that they contact. Also known colloquially as a “wedge”.

Pi: A constant, irrational number derived from the ratio of the circumference of a circle to its diameter. Pi always has the same value, regardless of the size of a circle. This value has been calculated to many trillions of digits, but 3.141 is fine for our purposes.


Important Comparisons

Below, there are several figures with descriptions that will help you remember the difference between the two important terms. Your teacher will very likely ask that you memorize these terms, so we've tried to help out a bit. Grab some Oreos and make your own chord.

Arc and Chord

Consider the difference between an arc and a chord. Think about the arc of a rainbow, give yourself a visual to remember. A chord is like a straight rope, pulled taut.

Figure 4
Figure 4: Remember the Difference Between "Arc" and "Chord"




Major and Minor Arcs

Make sure to understand the difference between major and minor arcs. A major arc is greater than half the circumference, while a minor arc is less than half the circumference. This should be easy to remember, as most people know the difference between major and minor.

Figure 5
Figure 5: Minor Arcs are less than half the circumference.
Figure 6
Figure 6: Major Arcs are more than half the circumference.




Central Angle

A central angle is an angle formed by two lines that meet at the centre, usually two radii, but not always. A chord or line segment the same length as a radius does not form a central angle when intersecting a radius unless the angle is at the centre.

In Figure 7 at intersection A there are four angles created by a radius and another line of radius length, none of these are central angles. Intersection B has three central angles. The length of the lines that create them are irrelevant.


Figure 7
Figure 7: The Central Angle is located at the Center of the Circle.



The Relationships Between Circles and Numbers

Look at Figure 8. We have a circle with its diameter and radius clearly labelled. We can deduce their lengths by calculating the difference between values on the plane.


Figure 8
Figure 8: Circle Located on a Coordinate Plane.

Now it's time for a little algebra to help us learn some of the relationships within the circle. If you haven't grabbed the Oreos, do so now. It's ok, the website will still be here when you get back from the kitchen.

We know that the diameter is twice the radius, d = 2r (therefore r = d/2). We know that pi (π) is the ratio of any circle’s circumference to its diameter, π = c/d (therefore c = πd and d = c/π). Imagine winding a piece of string around the circumference of a round cup for example. You will find that the diameter of that cup is the circumference length divided by π, or 3.141592653589793238462…

You will find out by doing some searching that Pi is an infinite number. For simplicity we will use a value of 3.14 for π here. Most modern calculators have a special button for π, but we’ll keep it simple because it’s the relationships we need to learn.

Take another look at Figure 8: See the space within the circle? This is the area. When working in two dimensions, like an architect on an blueprint or a builder on a wall or floor, we give this answer in units squared. When working in three dimensions like we will with spheres, cylinders and cubes, we’ll use units cubed instead of squared.

Calculating Area

The formula for an area is π*r2. We can calculate the area of the circle in Figure 8 as π*32, or π*9 which will be about 28. Using a calculator we get a value of 28.27.

Calculating Area of a Sector

To calculate the area of a sector, we need to know its central angle. Our calculation is a logical one: we use π*r2 to get the area of the circle then multiply that by the fraction our sector takes up.

The formula for this is:
π*r2 * (y/360)
where y is the number of degrees in the central angle of the sector.

Calculating Arc Length

If we know the circumference of a circle, we can deduce the length of any arc in that circle simply by multiplying the circumference length by the central angle the arc would have if it were a sector. It is exactly the same as the calculation above for sector area, except first we’re calculating circumference instead of area.

This is the formula we’ll use:
π*d * (y/360)
y is the number of degrees in a potential central angle connecting the two end points of the arc by radii.

Calculating Chord Length

Here we need to use a function called ‘Sine’ or SIN as it appears on many calculators. It’s frequently used in trigonometry. Here we will only look at one of its uses.

If we know the radius length and the central angle connected by two straight lines to the end points of the chord we can multiply half the central angle by the SIN function, then multiply that by the radius of the circle. This will yield half the length of the chord.

This is the formula we’ll use:
2 * (SIN (y/2) * r)
Where y is the number of degrees in a potential central angle connecting the two end points of the chord by radii, and r is the radius length.


Group Example

Let's look at Figure 9 together and work through the calculations below.

Figure 9
Figure 9: Group Problem.


To calculate the length of the chord in Figure 9, we first halve the central angle, y.
135.2/2 = 67.6

Then we enter SIN 67.6 into our calculator.
SIN 67.6 = 0.92

We’ll multiply this number by the radius, r, in this case 14.
0.92 * 14 = 12.88

This number is half the length of the chord. We’ll double it to get the entire length.
12.88 * 2 = 25.76




Sample Problem

Use the circle in Figure 10 as reference for the following Sample Problems. D is the central point, not the remaining angle. The radius is 11.6 Kilometres. Do the work on a scratch piece of paper and then check your answers using the '"show/ hide'" buttons.

Figure 10
Figure 10: Sample Problem - No Peeking!


1. The circumference.

ANSWER: π * 11.6 * 2 = 72.88 Km

2. The area.

ANSWER: π * 11.6 * 11.6 = 422.73 Km2

3. The diameter.

ANSWER: 11.6 * 2 = 23.20 Km

4. The length of chord AC.

ANSWER:
109/2 = 54.5
SIN 54.5 =0.8141
0.8141 * 11.6 = 9.4437
Twice this will equal the length of chord AC.
9.4437 * 2 = 18.8875 Km

5. Half of sector ABD’s area.

ANSWER:
π*11.62 * (67/360) = 78.6753
Halving this number will yield half of sector ABD’s area.
78.6753/2 = 39.3376 Km2

6. The area of major sector AED.

ANSWER:
First we’ll need to find the central angle:
Angle = 360 – 90 – 33 = 237°
Area = π * 11.62 * (237/360) = 278.2990 Km2

7. The area of Area 51 (made up of chord EF and arc EF).

ANSWER: We’ll need to get the area of sector EFD first. Because the central angle EDF is 90° we can simply multiply the area of the entire circle by ¼, because 90 is a quarter of 360. Then we need to subtract the area of triangle EDF from this number. We can calculate triangle EDF’s area by multiplying lines ED and FD, then halving that number. So our complete formula will look like this:
Area 51 = π * 11.62 * 0.25 – ½ * 11.62 Km2
Area = π * 11.62 * 0.25 = 105.6832
½ * 11.62 = 67.28
105.6832 – 67.28 = 38.4032 Km2