Start by posing the following questions to your students:
If each of the squares below is a unit square, what are the bounds for the area of the circle? What affect does increasing or decreasing the size of each square have on the accuracy of the bounds?
Solution: Count the number of squares which are completely shaded for a lower bound. Count the number of squares which are completely or partially shaded for an upper bound. As the size of each square increases, accuracy decreases, and vice-versa.
The idea for these questions came from here.
Next, students will see three ways of deriving the area formula, for circles, from the circumference formula. The first way is to inscribe a regular polygon in the circle, as seen here. The greater the number of sides, the more closely the area of the regular polygon will match the area of the circle. The second way is to cut slices, then rearrange into a rectangle, as seen here. The third way is to cut out concentric rings, then arrange into a triangle, as seen here. Then students will practice using the formula.
Area of a circle by Khan Academy
After that, students will learn how to find the area of composite figures involving semicircles.
Khan Academy includes only one of these derivations.
Watch these Khan Academy videos:
Next, give your students this challenge:
The perimeter of the square below is 8cm. Mentally determine the perimeter of the shaded region.
Here's the solution.
Conclude by leading this investigation:
Graphene Trampoline (logic & probability)
7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.