Unit 28 - Circumference and Pi

At the end of the lesson, the students are expected to:

• measure the circumference of various circular objects using various tools
• recognize and explain the consistent ratio between the circumference and diameter of circles, known as
• apply the formula C = π × d or C = 2 × π × r to calculate the circumference of a circle

Students will begin by measuring the circumference of smaller cylindrical objects, such as cans, and then move on to larger objects like hula hoops. They can use various instruments to measure the circumference, such as measuring tapes, meter sticks, or apps. Students should be instructed to record their measurements.

1. Ask the students to discuss their strategies for measuring the circumference of the circle. Be sure to highlight the strategy of rolling the objects to convert the circumference into a straight line (e.g https://www.geogebra.org/m/PVgpPjeq). This approach is crucial for accurate comparison.
2. Encourage students to examine their measurements and identify any patterns. At this stage, they should use their measuring instruments, such as tape measures or rulers, to compare measurements.
3. Have the students organize their data in a table. Ask them to look for relationships between the circumference and the diameter and confirm their observations through calculations. They should discover that the ratio of the circumference to the diameter is slightly more than 3, approximately 3.14. Ask them if this ratio is consistent across all circles.
4. Introduce the constant ratio that they have discovered, which is around 3.14, known as pi, and then introduce the symbol. Explain that this ratio is the same for all circles regardless of the size.

Students work in groups of 4–6 and form a human circle by standing shoulder-to-shoulder and extending their arms outward to connect with the students beside them, creating the circumference as accurately as possible. They then measure the distance around the human circle using a rope or measuring tape. Next, they measure the farthest distance between two students across the circle, representing the diameter. Students record both measurements and calculate the ratio circumference ÷ diameter. Students compare their results with those obtained from the circular objects in Activity 1. They may notice that the ratio for the human circle is less accurate or further from 3.14, because the students cannot form a perfectly circular shape and the distances between them may vary. Use this observation to reinforce that π is a constant for perfect circles, while real-world measurements can vary because of the accuracy of the shape and the measuring process.

Emphasize that for a circle of any size, the ratio of its circumference to its diameter is equal to Pi, which is about 3.14. This means that the circumference (C) of a circle is the product of Pi times the length of its diameter (d). That is, C = π × d Since the diameter is twice the radius, then d = 2 × r. So, we can also write C = 2 × π × r. As an exercise, let the students calculate the circumference of a circle given its radius/diameter.

• Science - collecting/recording data
• Technology - using apps to measure length
• Engineering - finding a way to measure the circumference, especially for large objects
• Physical Education - performing the measurement outside the classroom, rolling the hula hoop to draw a straight line
• Art - n/a
• Mathematics - performing simple calculations (circumference divided by diameter), finding the formula for circumference