Videos, examples, lessons, and solutions to help Grade 7 students learn how to give an informal derivation of the relationship between the circumference and area of a circle.
Circular Region (or Disk): Given a point C in the plane and a number r > 0, the circular region (or disk) with center C and radius r is the set of all points in the plane whose distance from the point C is less than or equal to r.
The boundary of a disk is a circle. The “area of a circle” refers to the area of the disk defined by the circle.
Solve the problem below individually. Explain your solution.
To find the formula for the area of a circle, cut a circle into 16 equal pieces.
Arrange the triangular wedges by alternating the “triangle” directions and sliding them together to make a “parallelogram”. Cut the triangle on the left side in half on the given line, and slide the outside half of the triangle to the other end of the parallelogram in order to create an approximate “rectangle”.
The circumference is 2πr, where the radius is “r”. Therefore, half of the circumference is πr.
What is the area of the “rectangle” using the side lengths above?
Are the areas of the rectangle and the circle the same?
Yes, since we just rearranged pieces of the circle to make the “rectangle,” the area of the “rectangle” and the area of the circle are approximately equal. Note that the more sections we cut the circle into, the closer the approximation. If the area of the rectangular shape and the circle are the same, what is the area of the circle?
Use the shaded square centimeter units to approximate the area of the circle.
What is the radius of the circle?
What would be a quicker method for determining the area of the circle other than counting all of the squares in the entire circle?
Using the diagram, how many squares did Michael use to cover one-fourth of the circle?
What is the area of the entire circle?
A sprinkler rotates in a circular pattern and sprays water over a distance of 12 feet. What is the area of the circular region covered by the sprinkler? Express your answer to the nearest square foot.
Draw a diagram to assist you in solving the problem. What does the distance of 12 feet represent in this problem? What information is needed to solve the problem?
Suzanne is making a circular table out of a square piece of wood. The radius of the circle that she is cutting is 3 feet. How much waste will she have for this project? Express your answer to the nearest square foot.
Draw a diagram to assist you in solving the problem. What does the distance of 3 feet represent in this problem?
What information is needed to solve the problem?
Does your solution answer the problem as stated?
Lesson 17 Problem Set Sample Solutions
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