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Lesson Plans and Worksheets for Geometry

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More Lessons for Geometry

Common Core For Geometry

Student Outcomes

- Students use inscribed and circumscribed polygons for a circle (or disk) of radius r and circumference C to show that the area of a circle is 1/2Cr or as it is usually written, πr
^{2}.

**Proving the Area of a Disk**

Classwork

**Opening Exercise**

The following image is of a regular hexagon inscribed in circle 𝐶 with radius 𝑟. Find a formula for the area of the hexagon in terms of the length of a side, 𝑠, and the distance from the center to a side.

**Example**

a. Begin to approximate the area of a circle using inscribed polygons.
How well does a square approximate the area of a disk? Create a sketch of 𝑃_{4}
(a regular polygon with 4 sides,
a square) in the following circle. Shade in the area of the disk that is not included in 𝑃_{4}.
b. Next, create a sketch of 𝑃_{8}
in the following circle.
c. Indicate which polygon has a greater area.
Area(𝑃_{4}) ____ Area(𝑃_{8})
d. Will the area of inscribed regular polygon 𝑃_{16} be greater or less than the area of 𝑃_{8}? Which is a better
approximation of the area of the disk?
e. We noticed that the area of 𝑃_{4} was less than the area of 𝑃_{8} and that the area of 𝑃_{8} was less than the area of
𝑃_{16}. In other words, Area(𝑃_{n}) ____ Area(𝑃_{2n}). Why is this true?
f. Now we will approximate the area of a disk using circumscribed (outer) polygons.
Now circumscribe, or draw a square on the outside of, the following circle such that each side of the square
intersects the circle at one point. We will denote each of our outer polygons with prime notation; we are
sketching 𝑃′_{4} here.
g. Create a sketch of 𝑃′_{8}.
h. Indicate which polygon has a greater area.
i. Which is a better approximation of the area of the circle, 𝑃′_{4} or 𝑃′_{8}? Explain why.
j. How will Area(𝑃′_{𝑛}) compare to Area(𝑃′_{2𝑛})? Explain.

**LIMIT (description)**: Given an infinite sequence of numbers, 𝑎1, 𝑎2, 𝑎3, …, to say that the limit of the sequence is 𝐴 means,
roughly speaking, that when the index 𝑛 is very large, then 𝑎𝑛 is very close to 𝐴. This is often denoted as, “As 𝑛 → ∞,𝑎𝑛 → 𝐴.”

**AREA OF A CIRCLE (description)**: The area of a circle is the limit of the areas of the inscribed regular polygons as the
number of sides of the polygons approaches infinity.

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