Test questions on the origin of the A = πr² formula via decomposition.

📚 Quick Study Guide

🔍 The area of a circle is given by the formula $A = \pi r^2$, where $A$ is the area and $r$ is the radius.

📐 Decomposition involves dividing the circle into smaller, manageable shapes, like sectors, which can then be rearranged to approximate a rectangle or parallelogram.

🧮 As the number of sectors increases (approaching infinity), the rearranged shape becomes closer to a perfect rectangle, making the area calculation more accurate.

💡 The length of this 'rectangle' approaches half the circumference of the circle ($\pi r$), and the width approaches the radius ($r$).

📝 Therefore, the area of the circle, approximated by the rectangle, is length × width = $(\pi r) × r = \pi r^2$.

Practice Quiz

1. What is the fundamental principle behind deriving the area of a circle formula using decomposition?
   1. A) Dividing the circle into equal triangles and summing their areas.
   2. B) Approximating the circle with a series of squares.
   3. C) Dividing the circle into sectors and rearranging them into a shape resembling a rectangle.
   4. D) Using calculus to integrate around the circle's circumference.
2. As the number of sectors in the decomposition method approaches infinity, what shape does the rearranged figure most closely resemble?
   1. A) A triangle.
   2. B) A square.
   3. C) A rectangle.
   4. D) A trapezoid.
3. In the decomposed and rearranged shape, what dimension corresponds to the radius ($r$) of the original circle?
   1. A) The length of the rectangle.
   2. B) The width of the rectangle.
   3. C) The diagonal of the rectangle.
   4. D) The perimeter of the rectangle.
4. In the rearranged shape, what dimension corresponds to half the circumference of the original circle?
   1. A) The width of the rectangle.
   2. B) The height of the rectangle.
   3. C) The area of the rectangle.
   4. D) The diagonal of the rectangle.
5. If a circle is decomposed into sectors and rearranged into a rectangle, what formula is used to calculate the area of this rectangle?
   1. A) $A = s^2$ (where $s$ is the side length).
   2. B) $A = \frac{1}{2}bh$ (where $b$ is the base and $h$ is the height).
   3. C) $A = lw$ (where $l$ is the length and $w$ is the width).
   4. D) $A = \pi r^2$ (where $r$ is the radius).
6. What is the area of a circle with radius $r$ after it has been decomposed and rearranged into an approximate rectangle?
   1. A) $A = 2\pi r$.
   2. B) $A = r^2$.
   3. C) $A = \pi r^2$.
   4. D) $A = 2r$.
7. Why does the decomposition method provide a good approximation for the area of a circle?
   1. A) Because the sectors perfectly form a square.
   2. B) Because the rearranged shape closely resembles a known geometric shape (rectangle).
   3. C) Because it simplifies the calculation of the circle's circumference.
   4. D) Because it eliminates the need for using $\pi$.