Proving the Area of a Disk

New York State Common Core Math Geometry, Module 3, Lesson 4

Worksheets 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, πr².

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.

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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 𝑃₄ (a regular polygon with 4 sides, a square) in the following circle. Shade in the area of the disk that is not included in 𝑃₄. b. Next, create a sketch of 𝑃₈ in the following circle. c. Indicate which polygon has a greater area. Area(𝑃₄) ____ Area(𝑃₈) d. Will the area of inscribed regular polygon 𝑃₁₆ be greater or less than the area of 𝑃₈? Which is a better approximation of the area of the disk? e. We noticed that the area of 𝑃₄ was less than the area of 𝑃₈ and that the area of 𝑃₈ was less than the area of 𝑃₁₆. 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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