Area of a Circle Lesson Plan

Warm-up

Students should write down what they already know about circles and share it with a partner. Encourage students to draw an example if needed. Students’ responses could vary, but students should activate their prior knowledge to see what will help them with finding the area of a circle.

Radius, diameter, and circumference

The goal of this warm up is to help ensure students recall the significance of pi with a circle’s circumference and that radius is always half of the diameter. Students should also be able to tell the difference between radius and diameter when given images.

Deriving the formula for area

To help students derive the formula for the area of a circle, you can let students know that the formula comes from rectangles. This video does a great job at showing the proof for the area of a circle.

Connection with rectangles

If a circle is cut infinitely many times, the slices could be put together to make a rectangle. Have students determine what the measurements of the rectangle are based on the circle. Ideally, students will recognize that the radius is like the height of the rectangle. Students may struggle with recognizing that the base of the rectangle is half of the circumference, so you may need to reiterate why that happens with students. When the base and height are multiplied, the area of the rectangle represents the area of the circle.

So the area of the rectangle would be radius `\cdot` half the circumference `= r \cdot  {2πr}/2 = πr^2`

Formula for area of a circle

Helping students understand the origin of the formula can help them make connections with what they already know; however, students will need to practice with the formula.

In terms of `\pi` and rounded

Give students a moment to ensure they write down the formula. To help students process the formula with a number, ask students to find the area of the circle if `r = 3`. This allows students to try and find the area. It also gives you an opportunity to explain the difference between finding the area in terms of `\pi` and writing an approximate value of the area.

Area given diameter

With the next example, students should recognize that they are given the diameter of the circle. Some students may ask, so it can help to clarify that when the value is in the middle of the circle, as shown, it indicates that it is the length of the diameter.

You may notice some of the following misconceptions:

• Not finding radius: Students may forget to find the radius from the diameter.
• Multiplying radius by `2` instead of squaring: Some students may accidentally do `4 \times 2` and get `8\pi`, so it is important to remind these students that `4^2` is like `4 \times 4`.
• Not writing area in terms of `\pi` and decimal form: Some students may only write their answer in terms of `\pi`, while others may only find the decimal approximation. Make sure students are comfortable writing their answers in either form. 

Real world application

To expose students to real-world problems, have them try this problem on their own or with a partner. 

Ideally, students will recognize that the length of the cable is the radius of the circle. Some students may think it is the diameter, so be mindful of that misconception. By this point, students should be more comfortable with finding the area of a circle.