CCSS.Math.Content.HSG-GPE.A.1
The Standard
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Common Core State Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students connect points on a circle to the center using horizontal and vertical distances. They form an equation from those distances and rewrite expanded equations to identify the center and radius.
What Mastery Looks Like
- Given a center and radius, students write the correct equation and explain what each part represents. From an expanded equation, they complete both squares, identify the center and radius, and verify their result.
Common Misconceptions
- Students often read (x - 3)^2 as a center coordinate of -3 and mistake r^2 for r. They may add a value to complete a square without adding the same value to both sides.
How to Assess It
- Exit ticket: Rewrite x^2 + y^2 - 6x + 8y - 11 = 0 in center-radius form. State the center and radius, and show both square-completion steps.
Lesson moves
Ways to Teach It
Pin a center on grid paper, use string cut to a set radius to trace a circle, then test plotted points with distances.
Ask why a circle centered at (3, -2) uses (x - 3)^2 and (y + 2)^2, then have students justify the signs.
Run a card sort matching expanded equations, center-radius forms, centers and radii, and graphs; require one square-completion calculation for each match.
Map a sprinkler at (4, 6) with a 5-meter reach, then decide whether three marked garden beds receive water.
Free download
Printable HSG-GPE.A.1 Worksheet
A ready-to-print activity worksheet aligned to CCSS.Math.Content.HSG-GPE.A.1, with an answer key for the teacher on its own page. No account needed.
PDF, US Letter, prints cleanly in black and white.
Download the worksheetLearning progression
Where This Standard Sits
Before This Standard
If students are struggling here, check these first.
- CCSS.Math.Content.8.G.B.8
Circle equations come from setting the coordinate distance from center to point equal to the radius.
- CCSS.Math.Content.HSA-REI.B.4a
Completing the square for quadratics carries over to rewriting circle equations into center-radius form.
- CCSS.Math.Content.HSA-SSE.A.2
Recognizing quadratic structure supports completing the square to rewrite a circle equation in center-radius form.
What This Unlocks
Mastery here sets students up for these next.
- CCSS.Math.Content.HSG-GPE.A.2
Students reuse distance relationships and coordinate algebra from circle equations to express points equidistant from a focus and directrix.
- CCSS.Math.Content.HSG-GPE.A.3
Circle equations practice distance-based loci and completing squares, which support deriving and recognizing ellipse and hyperbola equations.
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