Overview of Section 4.5 Circumference and Area of a Circle
The text presents an intuitive and informal approach to the topics of circumference and area of a circle. Definitions are avoided because, essentially, formal definition requires the use of limits. We begin with a proposition from Euclid that
"the areas of any two circles are to each other as the squares of their diameters"
This is very consistent with Theorem 4.14 and its corollaries. The ratio of the areas of similar polygonal figures are proportional to the square of the ratio of corresponding linear measures. Viewing the circle as a limiting case of an n-sided polygon then the extension to circles is reasonable.
The equation for Euclid's proposition is
This implies that the ration of the areas of any circle to the square of its diameter is a constant:
There is supposed to be a proof explored in Problem set 4.6 but THERE IS NO PROBLEM SET 4.6!!! For the moment we can designate that constant
On the other hand, it is also known that the circumferences of two circles are related to each other as the corresponding diameters
and consequently we can write
This implies that the ratio of the circumference to its diameter is a constant and we write
Of course the constants k and k' are equal and the value of the constant is .
A Plausible argument for deriving the formula for the area of a circle from the formula for its circumference.
The gist of this argument is that if we express the area of an n-sided regular polygon, as n gets large the area of the polygon approaches the area of the circle. We can have a formula for the area of the polygon in terms of its perimeter by looking at a sector of the inscribed n-gon as follows:
Archimedes's Estimation of
The strategy attributed to Archimedes for estimating the value of is summarized by examining a circumscribed n-gon about a unit circle and an inscribed n-gon in the unit circle. As the number of sides gets large, the areas of the two polygons get closer and closer together. Therefore we can use the difference of the two areas in the limit to approximate .
Ben Smith's Trapezoid Problem
The problem from Ben Smith was shared by e-mail. Some of you have provided a solution and the GSP files are linked here.
Given an isosceles trapezoid with known lengths a and b, drawing of the trapezoid with the diagonals and the arithmetic mean. So, the mean is split into 3 parts by the diagonals forming several similar triangles. The question asks to find the length of the smaller middle section of the arithmetic mean. However, the problem does not give any values for height. Does this problem have solution?
Does this problem have a solution? Why was Ben led to ask this question? He seemed to recognise the key geometric ideas.
One element of the solution, if it exists, would be to show that the result is independent of the height of the trapezoid.
GSP File with my solution
GSP File with solution by Victor
GSP File with solution by Allyson
GSP File with Solution by Jackie
All of us have presented essentially the same solution. Are there alternative solutions? We made no use of the 'several similar triangles' Ben Smith mentioned. The diagonals of a trapezoid intersect at a point corresponding to the harmonic mean. Could similar triangles or something about harmonic and arithmetic means be used?
Consider this alternative.
Construct CX parallel to DA. XB = b - a
AXCD is a parallelogram with G as the midpoint of diagonal AC
XD is a diagonal of AXCD and so G is a midpoint of XD
H is a midpoint of DB.
Therefore by the triangle midsegment theorem, .
Trigonometry Problem Presented in Class
1. Use the sublime triangle to get values for sin 18 and cos 36
See GSP file for solution by Kathy Radford
See GSP file for other solutions
2. Use trigonometric equation
If P is a point on the secant AB outside the circle and PT a tangent to the circle such that PT = AB. Locate C inside the circle such that PC = PT. See GSP File if needed.
A Problem using a 3-4-5 Right Triangle
Five Circular Discs Problem
Five unit discs are placed with centers on the vertices of a regular pentagon and all passing through the center of the Pentagon.
What is the radius of the largest circular region that would be covered by the discs? OA = ?
Note: The discs have a radius of 1 and their centers are on the vertices of a pentagon. Thus this pentagon is inscribed in a circle with unit radius.
The intersections of the discs will form a larger regular pentagon and it is the radius of the circumscribing circle for the larger pentagon that is sought by the problem.
Problem Set 4.5