Education Technology

Exploring the Equation of a Circle

Updated on 06/12/2015

Activity Overview

Explore right triangles and the Pythagorean Theorem to develop the equation of a circle.

Objectives

  • Students will understand the definition of a circle as a set of all points that are equidistant from a given point.
  • Students will understand that the coordinates of a point on a circle must satisfy the equation of that circle.
  • Students will relate the Pythagorean Theorem and Distance Formula to the equation of a circle.
  • Given the equation of a circle (x – h)2 + (y – k)2 = r 2, students will identify the radius r and center (h, k).

Vocabulary

  • Pythagorean Theorem
  • Distance Formula
  • Radius

About the Lesson

This lesson involves plotting points that are a fixed distance from the origin, dilating a circle entered on the origin, translating a circle away from the origin, and dilating and translating a circle while tracing a point along its circumference. As a result students will:

  • Visualize the definition of a circle.
  • Visualize the relationship between the radius and the hypotenuse of a right triangle.
  • Observe the consequence of this manipulation on the equation of the circle.
  • Infer the relationship between the equation of a circle and the Pythagorean Theorem.
  • Infer the relationship between the equation of a circle and the Distance Formula.
  • Identify the radius r and center (h, k) of the circle (xh)2 + (y k)2 = r 2.
  • Deduce that the coordinates of a point on the circle must satisfy the equation of that circle.