Equations of a Circle

Rating:

- (2)
- (0)
- (0)
- (0)
- (1)

Description:

Show how the Standard Equation of a Circle is derived from the Distance Formula.
Present how to write the standard form of a circle’s equation given the central point (h, k) and a radius.
Show how to find the center (h, k) and radius of a circle when given an equation in standard form, and then show how to graph the circle.
Introduce the General Form of the Equation of a Circle [(x – h)² + (y – k)² = r²], and show how it can be derived from the Standard Form.
Convert the general form of a circle’s equation to standard form.

This packet should help a learner to understand how to use the equations of a circle to gather information and graph a circle.

(more)

Try Our College Algebra Course. For FREE.

Sophia’s self-paced online courses are a great way to save time and money as you earn credits eligible for transfer to over 2,000 colleges and universities.*

Begin Free Trial

No credit card required

28 Sophia partners guarantee credit transfer.

253 Institutions have accepted or given pre-approval for credit transfer.

* The American Council on Education's College Credit Recommendation Service (ACE Credit®) has evaluated and recommended college credit for 21 of Sophia’s online courses. More than 2,000 colleges and universities consider ACE CREDIT recommendations in determining the applicability to their course and degree programs.

Tutorial

Introduction to Circles

This video presents the standard form of the equation of a circle and shows how it relates to the graph of the circle.

open player in a new window

Source: David on Guaranteach

Derivation of the Standard Equation of a Circle

Using the distance formula, it's fairly easy to see where our standard equation of a circle comes from. We want to find all the points that are a distance r from the center point - that's the definition of a circle. Call the center point (h,k), and choose an arbitrary point (x,y) on the circle, like so:

For every (x,y), the distance formula tells us that the distance between (h,k) and (x,y) is $square root of left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared end root$ . We know that we want this distance to be r, so we have $r space equals space square root of left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared end root$ . By squaring both sides, we get $r squared equals left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared$ , which is the standard form of the equation of a circle.