4.5.3 Circumference and Area of Circles

Calculate circumference and area of circles using formulas C = 2πr and A = πr².

定义

A circle is a two-dimensional geometric shape consisting of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius, denoted as \(r\). The circumference of a circle is the total distance around the circle, while the area of a circle is the measure of the space enclosed within the circle. Both circumference and area are fundamental properties that depend on the radius of the circle and the mathematical constant \(\pi\) (pi), which is approximately 3.14159.

核心公式

• \(C = 2\pi r\)
• \(C = \pi d\)
• \(A = \pi r^2\)
• \(d = 2r\)
• \(r = \frac{C}{2\pi}\)

易错点

• ⚠️ Confusing radius and diameter: Students often use the diameter instead of the radius in the area formula, calculating \(A = \pi d^2\) instead of \(A = \pi r^2\), which results in an answer that is 4 times too large.
• ⚠️ Using the wrong formula for circumference: Applying the area formula to find circumference or vice versa, such as using \(C = \pi r^2\) instead of \(C = 2\pi r\).
• ⚠️ Forgetting to include \(\pi\) in the final answer: Some students calculate the numerical coefficient correctly but forget to multiply by \(\pi\), giving answers like 25 instead of \(25\pi\) for the area of a circle with radius 5.
• ⚠️ Incorrectly simplifying expressions with \(\pi\): When given circumference or area in terms of \(\pi\), students may make algebraic errors when solving for radius, such as dividing by \(2\pi\) incorrectly when rearranging \(C = 2\pi r\).