Why the Area of a Circle is \(A=\pi r^2\)

🔑 Key Concepts

  • A circle can be divided into many narrow sectors and rearranged.
  • Alternating the sectors creates a figure that approaches a rectangle.
  • The height of the rearranged figure is the radius \(r\).
  • The top and bottom together equal the full circumference \(2\pi r\).
  • Each horizontal side is therefore half the circumference, or \(\pi r\).
  • Multiplying the base and height gives \(A=(\pi r)(r)=\pi r^2\).

✏️ Worked Example

🧠 Math Vocabulary

  • Circle: The set of all points in a plane that are the same distance from a fixed center point.
  • Radius: A segment from the center of a circle to a point on the circle.
  • Circumference: The distance around a circle.
  • Sector: A region of a circle bounded by two radii and part of the circle’s curved edge.
  • Rearrange: To move pieces into a different position without changing their total area.
  • Approximate: To give a value or shape that is close to an exact value or shape.
  • Pi: The ratio of a circle’s circumference to its diameter, represented by \(\pi\).
  • Area: The amount of two-dimensional space inside a figure.
  • Derivation: A sequence of mathematical reasoning used to explain where a formula comes from.

💡 Main Idea

The formula \(A=\pi r^2\) can be understood by dividing a circle into many narrow sectors and rearranging them into a nearly rectangular figure. Half of the circle’s circumference forms the base, so the base is \(\pi r\). The radius forms the height. Multiplying the base and height gives \((\pi r)(r)=\pi r^2\).

📚 What You Should Already Know

You should understand the meaning of radius, circumference, and area. You should also know that the circumference of a circle is \(2\pi r\) and that the area of a rectangle is found by multiplying its base and height.

🚀 What Comes Next

After understanding why the formula works, students can calculate circle areas from radii or diameters, compare the areas of circles, find missing dimensions, and solve composite-area problems involving circles, semicircles, and shaded circular regions.

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