Lesson 11

Approximating Pi

  • Let’s approximate the value of pi.

11.1: More Sides

Square BCDE inside of circle with center A, radius = 1. Shaded region outside of square. Distance between vertices of square = the square root of 2. 
Hexagon BCDEFG inside of circle with center A, radius = 1. Shaded region outside of hexagon. Distance between vertices of hexagon = 1.

Calculate the area of the shaded regions.

11.2: N Sides

The applet shows a regular \(n\)-sided polygon inscribed in a circle. 

Come up with a general formula for the perimeter of the polygon in terms of \(n\).  Explain or show your reasoning.

11.3: So Many Sides

Let's use the expression you came up with to approximate the value of \(\pi\).

  1. How close is the approximation when \(n=6\)?
  2. How close is the approximation when \(n=10\)?
  3. How close is the approximation when \(n=20\)?
  4. How close is the approximation when \(n=50\)?
  5. What value of \(n\) approximates the value of \(\pi\) to the thousandths place?


Describe how to find the area of an \(n\)-gon with side length \(s\). Then write an expression that will give the area.

Summary

It's easier to work with polygons than with circles because we can decompose polygons into simple shapes such as triangles. We can use polygons to figure out things about circles. For example, we know how to calculate the area of regular polygons inscribed in a circle of radius 1.

Pentagon inscribed in circle. 2 segments, length 1, from center to vertices form triangle. Dotted line, h, from center to triangle base forms 2 right triangles. Other right triangle leg is x. Angle between h and 1 is theta.

To find the area of this regular pentagon, let's find the area of one triangle and then multiply by 5. Drawing in the altitude creates a right triangle, so we can use trigonometry to calculate the lengths of both \(x\) and \(h\). To find \(\theta\) use the fact that a full rotation is \(360^\circ\) and that in an isosceles triangle the altitude is also an angle bisector. So \(\theta=360 \div 10\). \(\sin(36)=\frac{x}{1}\) so \(x\) is about 0.59 units. \(\cos(36)=\frac{h}{1}\) so \(h\) is about 0.81 units. The area of the isosceles triangle is about 0.48 square units and the area of the pentagon is 5 times that, or about 2.4 square units.

That's not very close to the area of the circle, but if we add more and more sides to the regular polygon, its area gets closer and closer to covering the entire circle.

Glossary Entries

  • arccosine

    The arccosine of a number between 0 and 1 is the acute angle whose cosine is that number.

  • arcsine

    The arcsine of a number between 0 and 1 is the acute angle whose sine is that number.

  • arctangent

    The arctangent of a positive number is the acute angle whose tangent is that number.

  • cosine

    The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. In the diagram, \(\cos(x)=\frac{b}{c}\).

  • sine

    The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse. In the diagram, \(\sin(x) = \frac{a}{c}.\)

  • tangent

    The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. In the diagram, \(\tan(x) = \frac{a}{b}.\)

  • trigonometric ratio

    Sine, cosine, and tangent are called trigonometric ratios.