Equation of a circle. Game on.

Equation of a circle is a grade 10 math skill aligned to Common Core standard HSG.GPE.A.1: 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. Below are 8 practice questions with answers and step-by-step explanations, drawn from the 20 equation of a circle problems our math games drill.

CCSS HSG.GPE.A.120 questions in the bank
Kickoff

8 plays. No signup.

Warm-upeasy

A circle in the coordinate plane has equation (x−3)2+(y+1)2=16.(x - 3)^{2} + (y + 1)^{2} = 16\textsf{.} What is the center of the circle?

Standard form is (x−h)2+(y−k)2=r2.(x - h)^{2} + (y - k)^{2} = r^{2}\textsf{.} Here h=3h = 3 and k=−1,k = -1\textsf{,} so the center is (3,−1).(3, -1)\textsf{.}

Mid-gameeasy

A circle in the coordinate plane has equation (x+8)2+(y−3)2=49.(x + 8)^{2} + (y - 3)^{2} = 49\textsf{.} What is the xx-coordinate of the center of the circle?

(x+8)2=(x−(−8))2,(x + 8)^{2} = (x - (-8))^{2}\textsf{,} so the center's xx-coordinate is −8.-8\textsf{.}

Mid-gameeasy

A circle in the coordinate plane has equation (x+1)2+(y−7)2=4.(x + 1)^{2} + (y - 7)^{2} = 4\textsf{.} What is the yy-coordinate of the center of the circle?

(y−7)2(y - 7)^{2} means k=7,k = 7\textsf{,} so the center is (−1,7).(-1, 7)\textsf{.}

Mid-gameeasy

A circle in the coordinate plane has equation (x−2)2+(y−7)2=36.(x - 2)^{2} + (y - 7)^{2} = 36\textsf{.} What is the measure of the diameter of the circle?

The radius measures 66 units, so the diameter measures 1212 units.

Mid-gameeasy

A circle in the coordinate plane has equation (x−1)2+(y−4)2=81.(x - 1)^{2} + (y - 4)^{2} = 81\textsf{.} What is the value of r2?r^{2}\textsf{?}

In standard form the right side equals r2,r^{2}\textsf{,} so r2=81.r^{2} = 81\textsf{.}

Mid-gameeasy

A circle in the coordinate plane has equation (x−3)2+(y+2)2=25.(x - 3)^{2} + (y + 2)^{2} = 25\textsf{.} What is the measure of the radius of the circle?

The right side is r2=25,r^{2} = 25\textsf{,} so the radius measures 55 units.

Mid-gameeasy

A circle in the coordinate plane is centered at the origin and passes through the point (6,8).(6, 8)\textsf{.} Which equation represents the circle?

The radius is the distance from (0,0)(0, 0) to (6,8):(6, 8)\textsf{:} 62+82=100=10\sqrt{6^2 + 8^2} = \sqrt{100} = 10 units. Standard form is (x−h)2+(y−k)2=r2,(x - h)^2 + (y - k)^2 = r^2\textsf{,} so x2+y2=102=100.x^2 + y^2 = 10^2 = 100\textsf{.}

Buzzer beatereasy

A circle in the coordinate plane has equation (x−3)2+(y+1)2=16.(x - 3)^{2} + (y + 1)^{2} = 16\textsf{.} What is the measure of the radius of the circle?

The right side is r2=16,r^{2} = 16\textsf{,} so the radius measures 44 units.

Overtime

12 more. Played, not assigned.

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Tests that cover equation of a circle:
NAEPSATACTNY State MathNC EOG/EOC
Reporting domains for this skill
SAT All grades: Geometry and TrigonometryACT All grades: Preparing for Higher MathNAEP Grade 12: Measurement and GeometryACT Aspire All grades: Preparing for Higher MathATLAS Geometry: CirclesATLAS Geometry: Expressing Geometric Properties with Equations
More Grade 10 skills
Midpoint formulaPossible coordinates on a circle (SAT)Perimeter and arc lengthRight triangle trig from two ratiosCircle equation translations (ACT/SAT/PSAT 8/9/10)Circle diameter endpoints to radius (SAT)Special right trianglesTwo-event conditional probability word problems (SAT)All Grade 10 practice →

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