Annenberg – Circles and Pi

Sorry for the vast amount of notes, but I never studied the meaning behind these formulas when I was in grade school, so I found this lesson really helpful!

  • Perimeter — or distance around — is a measurable property of simple, closed curves and shapes. When the figure is a circle, we use the term circumference instead of perimeter. Because the perimeter of an object is a length, we need to measure using units of length such as centimeters, decimeters, meters, inches, feet, etc.
  • A diameter is a chord — a line segment joining two points on the arc of a circle — that passes through the center of the circle. Diameter also refers to the distance between two points on the circle, measured through the center.
  • The three designs below show a circle between a regular hexagon and a square:

Screen shot 2013-07-29 at 6.12.37 PM

Screen shot 2013-07-29 at 6.13.55 PM

Relationships of the designs above:

  • The measurements stay in scale. In all three, the diagonal of the hexagon is twice the length of the hexagon’s side. Also, as we move from one design to the next, the length of each side of the inscribed hexagon increases by 1; the length of each side of the inscribed hexagon is equal to the radius of the circle (as shown by the inscribed equilateral triangles).  The length of each side of the square is the same as the diameter of the circle inscribed within; the ratio of the length of the diameter of the circle to the length of one side of the hexagon is 2/1 for all three designs.
  • The perimeter of the hexagon is three times the diameter of the circle, and the perimeter of the square is four times the diameter of the circle.
  • The circumference of the circle is between the perimeter of the square and the perimeter of the hexagon, but closer to the hexagon’s perimeter.

Circles and Circumference:

  • All circles have one trait in common: The ratio of circumference to diameter is a constant value, which is a little more than 3.  Pi is an irrational number.  Its decimal part continues forever without repeating. As of 1997,  pi had been extended to 51 billion decimal places (using a computer)! Your calculator has a special key for pi, but this is only an approximate value.
  • We’ve seen that  pi = C/d, where C is the circumference and d is the diameter of a circle. We can multiply both sides of the equation by d to get a new equation as C = (pi) * d. Because the diameter of a circle is always twice its radius, we can write the new equation as C = pi • 2r
  • Since pi is irrational, One or the other (the circumference or the diameter) may be rational, but not both. If they were both rational, their ratio (which is pi ) would also have to be rational, which it is not. A circle may have a diameter of exactly 12 cm with an irrational circumference, or a circumference of exactly 100 m with an irrational diameter.  They can however, both be irrational.
  • Two forms of the standard equation that shows the relationship between circumference and diameter: C = (pi) • d and  pi = C/d
  • Since pi is an irrational number, the exact circumference can only be expressed using the symbol for pi. Sometimes, however, we want to solve a real problem and find an approximate value for a circumference. In that case, we must use one of the approximations for pi . Inexactness may also occur when determining a numerical value for circumference (or diameter) because of measurement error.

Areas of a Circle:

Transforming a circle into a crude parallelogram:

Screen shot 2013-07-29 at 6.32.14 PM

The scalloped base of the figure is formed by arcs of the circle.

The length of the base is one-half the circle’s circumference, since the entire circumference comprises the scalloped edges that run along the top and bottom of the figure, and exactly half of it appears on each side. The base length is C/2.

Since the circumference is 2 • pi  • r, where r is the radius, the base is half of this. The base length is  pi • r

As the number of wedges increases, each wedge becomes a nearly vertical piece. The base length becomes closer and closer to a straight line of length  pi • r (or half the circumference), while the height is equal to r. The area of such a rectangle is  pi • r • r, or  pi • r2.

The area formula of a circle is A =  pi • r2.

If r is the radius of a circle, then r2 is a square with sides of length r. Examine the circles below. A portion of each circle is covered by a shaded square. We can call each of these squares a radius square.

Screen shot 2013-07-29 at 6.35.26 PM

Screen shot 2013-07-29 at 6.35.59 PM

 

In each case, it takes a little more than three radius squares to form the circle. If using approximations, it should always take around 3.14 of the squares to cover the circle.  The best estimate of the area of any circle in radius squares is somewhere between 3.1 and 3.2, which we know is roughly the value of pi.

Think about a circle with a radius equal to 1 (r = 1). The circumference and the area of this circle are as follows:

C = 2 • 1 • pi = 2 (pi)

A = 12 • pi = pi

Now double the radius to 2 units (r = 2). The circumference and the area of the new circle are as follows:

C = 2 • 2 • pi  = 4(pi)

A = 22 • pi = 4 (pi)

The circumference of the new circle doubled, but the area is multiplied by a factor of 4 (the square of the scale factor). You can replace the 1 with any other number, or with a variable r, to see that this relationship will always hold.

 

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