Geometry of Circles

By using a MIRA, students determine the relationship between radius, diameter, circumference, and area of a circle. This is a great hands on activity to develop the relationship of various aspects of the circle.

Standards & Objectives

Academic standards
CCSS.Math.Content.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge...
CCSS.Math.Content.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures....
CCSS.Math.Content.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...
CCSS.Math.Content.7.G.B.6
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles,...
CCSS.Math.Content.8.G.C.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
GLE 0606.1.3
Develop independent reasoning to communicate mathematical ideas and derive algorithms and/or formulas.
GLE 0606.4.3
Develop and use formulas to determine the circumference and area of circles, and the area of trapezoids, and develop strategies to find the area of composite shapes.
GLE 0606.4.4
Develop and use formulas for surface area and volume of 3-dimensional figures.
SPI 0606.4.4
Calculate with circumferences and areas of circles.
SPI 0606.4.5
Determine the surface area and volume of prisms, pyramids and cylinders.
SPI 0606.4.6
Given the volume of a cone/pyramid, find the volume of the related cylinder/prism or vice versa.
TSS.Math.6.G.A.2
Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and...
TSS.Math.6.G.A.4
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these...
TSS.Math.7.G.B.3
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...
TSS.Math.7.G.B.5
Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles,...
TSS.Math.8.G.C.7
Know and understand the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.
 
Alignment of this item to academic standards is based on recommendations from content creators, resource curators, and visitors to this website. It is the responsibility of each educator to verify that the materials are appropriate for your content area, aligned to current academic standards, and will be beneficial to your specific students.
 
Learning objectives: 

Learning Objectives:

Students will:

  • Construct circles, and identify the diameters and centers of those circles.
  • Understand the relationship between diameter and circumference.
  • Understand the relationship between radius and the area.
Essential and guiding questions: 

Questions for Students:

  • How do you know if a chord of a circle is also a diameter?
  • How is the diameter of a circle used to find its circumference?
  • How is the radius of a circle used to find its area?

Lesson Variations

Blooms taxonomy level: 
Understanding
Extension suggestions: 

Extensions:

  • To develop understanding of the area of a circle, have pairs of students cut up a paper plate using lines of symmetry through the center, just as one slices a pizza. Rearrange the slices as shown below. Students will realize that this configuration almost looks like a rectangle! How would this "rectangle" help in finding the area of a circle? [The width of the rectangle is equal to the radius of the original circle. The length of the rectangle is half of the circumference, since the entire circumference is both on the top and bottom. Therefore, the area is equal to the radius times half the circumference, or A = ½Cr. Because C = =πd and d = 2r, this formula becomes the more familiar A = πr2.]
  • Allow students to use Geometer’s Sketchpad or other geometry software to create the constructions described in this lesson.
  • Research how hat sizes were determined! Or, check out the web site of a company that makes and sells hats, and you might find a table like the one below. What is the relationship between men’s head measurement (in inches) and American hat sizes? Have students measure the circumference of their head, and divide it by π — the result is their hat size.
  •  If students were to plot these points in a scatterplot, one reasonable line of best fit is y = 3.14x, indicating that the y‑value (head circumference) is approximately π times the x‑value (hat size).

Helpful Hints

Materials:

  • MIRATM Geometry Tool 
  • Compass 
  • Geometer's Sketchpad software program on the computer (optional) 

References

Contributors: