Circumference and Area of Circles - Sixth Grade Mathematics

The circumference and area of a circle are governed by the fundamental mathematical constant pi, which defines the universal ratio of a circle's perimeter to its diameter.

The Discovery of Pi (π)

In ancient mathematics, scholars discovered that dividing the distance around any circle (circumference C) by its width through the center (diameter d) yields the exact same ratio regardless of the circle's size.

The Universal Constant Pi:
               Circumference C
Ratio = pi = -------------------  approx. 3.14159265...
                 Diameter d

Key Approximations used in Grade 6:
- Decimal approximation: pi approx. 3.14
- Fraction approximation: pi approx. 22 / 7 (especially convenient when d or r is a multiple of 7)

Because pi is an irrational number, its decimal digits continue infinitely without repeating.

Circumference of a Circle

Circumference is the linear distance around the outside edge of a circle.

Circumference Formulas:
C = pi * d          (when diameter d is given)
C = 2 * pi * r      (when radius r is given, since d = 2r)

Example 1: A bicycle wheel has a diameter of 28 inches. Find its circumference using pi approx. 22/7.
           C = pi * d = (22 / 7) * 28 = 22 * 4 = 88 inches.

Example 2: A circular swimming pool has a radius of 5 meters. Find its circumference using pi approx. 3.14.
           C = 2 * pi * r = 2 * 3.14 * 5 = 10 * 3.14 = 31.4 meters.

Deriving the Area of a Circle

To understand why the area formula works, imagine cutting a circle into many equal pie slices (sectors) and rearranging them head-to-toe.

Unfolding Circle Sectors into a Parallelogram:
Cut circle into 16 sectors:
         /\/\/\/\/\/\/\/\     <- Half of circumference = (1/2) * (2 * pi * r) = pi * r
        +----------------+
        |                |  <- Radius r
        +----------------+
         \/\/\/\/\/\/\/\/     <- Other half of circumference = pi * r

As the number of wedges increases, the shape becomes a perfect RECTANGLE!
Base of rectangle = pi * r
Height of rectangle = r
Area = base * height = (pi * r) * r = pi * r^2!

Circle Area Formula:
Area = pi * r^2
Order of Operations Reminder: Square the radius FIRST, then multiply by pi!

Example: A circular pizza has a radius of 6 inches. Find its area using pi approx. 3.14.
         A = pi * r^2 = 3.14 * 6^2 = 3.14 * 36 = 113.04 square inches.
         If diameter had been given as 12 inches, find radius first: r = 12 / 2 = 6 inches!

Semicircles and Quarter Circles

Fractional portions of circles require scaling the area and perimeter accordingly: Area of Semicircle = (1 / 2) * pi * r^2. Area of Quarter Circle = (1 / 4) * pi * r^2. Perimeter of a Semicircle = (1 / 2) * (pi * d) + d (remember to add the straight diameter baseline!).

Exercises

Exercise 1. Calculate the circumference and area of each circle (use pi approx. 3.14): (a) Radius = 10 cm; (b) Diameter = 14 meters; (c) Radius = 2.5 inches.

Exercise 2. Use pi approx. 22/7 to find: (a) The circumference of a circular track with diameter 35 meters; (b) The area of a circular rug with radius 7 feet.

Exercise 3. A circular fountain has a circumference of 50.24 feet. (a) What is the diameter of the fountain? (b) What is its radius? (c) What is the area covered by the fountain? (Use pi approx. 3.14).

Exercise 4. A lawn sprinkler sprays water in a circle of radius 15 feet. What is the total square footage of lawn watered by the sprinkler?

Solutions

Solution to Exercise 1. For (a), C = 2 * 3.14 * 10 = 62.8 cm; A = 3.14 * 10^2 = 3.14 * 100 = 314 cm^2. For (b), r = 14 / 2 = 7 meters; C = 3.14 * 14 = 43.96 meters; A = 3.14 * 7^2 = 3.14 * 49 = 153.86 m^2. For (c), C = 2 * 3.14 * 2.5 = 5 * 3.14 = 15.7 inches; A = 3.14 * 2.5^2 = 3.14 * 6.25 = 19.625 in.^2.

Solution to Exercise 2. For (a), C = pi * d = (22 / 7) * 35 = 22 * 5 = 110 meters. For (b), A = pi * r^2 = (22 / 7) * 7^2 = (22 / 7) * 49 = 22 * 7 = 154 ft^2.

Solution to Exercise 3. For (a), C = pi * d, so 50.24 = 3.14 * d. d = 50.24 / 3.14 = 16 feet. For (b), r = d / 2 = 16 / 2 = 8 feet. For (c), A = pi * r^2 = 3.14 * 8^2 = 3.14 * 64 = 200.96 ft^2.

Solution to Exercise 4. Area = pi * r^2 = 3.14 * 15^2 = 3.14 * 225 = 706.5 square feet.