Lesson 4

Applying Circumference

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Lesson Narrative

In this lesson, students use the equation \(C=\pi d\) to solve problems in a variety of contexts. They compute the circumference of circles and parts of circles given diameter or radius, and vice versa. Students develop flexibility using the relationships between diameter, radius, and circumference rather than memorizing a variety of formulas. Understanding the equation \(C = 2 \pi r\) will help with the transition to the study of area in future lessons.

Students think strategically about how to decompose and recompose complex shapes (MP7) and need to choose an appropriate level of precision for \(\pi\) and for their final calculations (MP6).

Learning Goals

Teacher Facing

  • Apply understanding of circumference to calculate the perimeter of a shape that includes circular parts, and explain (orally and in writing) the solution method.
  • Compare and contrast (orally) values for the same measurements that were calculated using different approximations for $\pi$.
  • Explain (orally) how to calculate the radius, diameter, or circumference of a circle, given one of these three measurements.

Student Facing

Let’s use \(\pi\) to solve problems.

Required Materials

Learning Targets

Student Facing

  • I can choose an approximation for $\pi$ based on the situation or problem.
  • If I know the radius, diameter, or circumference of a circle, I can find the other two.

CCSS Standards

Addressing

Glossary Entries

  • circle

    A circle is made out of all the points that are the same distance from a given point.

    For example, every point on this circle is 5 cm away from point \(A\), which is the center of the circle.

    A circle with points A, B, C, D, E, F
  • circumference

    The circumference of a circle is the distance around the circle. If you imagine the circle as a piece of string, it is the length of the string. If the circle has radius \(r\) then the circumference is \(2\pi r\).

    The circumference of a circle of radius 3 is \(2 \boldcdot \pi \boldcdot 3\), which is \(6\pi\), or about 18.85. 

  • diameter

    A diameter is a line segment that goes from one edge of a circle to the other and passes through the center. A diameter can go in any direction. Every diameter of the circle is the same length. We also use the word diameter to mean the length of this segment.

    A circle with its diameter labeled
  • pi ($\pi$)

    There is a proportional relationship between the diameter and circumference of any circle. The constant of proportionality is pi. The symbol for pi is \(\pi\).

    We can represent this relationship with the equation \(C=\pi d\), where \(C\) represents the circumference and \(d\) represents the diameter.

    Some approximations for \(\pi\) are \(\frac{22}{7}\), 3.14, and 3.14159.

    a graph in the coordinate plane
  • radius

    A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.

    For example, \(r\) is the radius of this circle with center \(O\).

    a circle with a labeled radius