← Back to the Equation Of A Circle worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: G-GPE.A.1 · Level: Grade 10 Geometry
Problem 2 Centered at the Origin
x² + y² = 11² = 121.
x2 + y2 = 121
Problem 3 Read Center and Radius
(x + 2) = (x − (−2)) and (y + 8) = (y − (−8)), so h = −2 and k = −8. The radius is √64 = 8.
Center (−2, −8), radius 8
Problem 4 Center on the x-Axis
(x + 5)² + (y − 0)² = (√10)² = 10.
(x + 5)² + y² = 10
Problem 5 Equation from a Graph
The center is (2, 1). The point (6, 1) is 4 units to the right, so r = 4.
(x − 2)² + (y − 1)² = 16
Problem 6 Radius from a Point
r² = (1 − (−3))² + (7 − 2)² = 16 + 25 = 41.
(x + 3)² + (y − 2)² = 41
Problem 7 Equation from a Diameter
Center = midpoint = (−4 + 62, 3 + (−1)2) = (1, 1).
r² = (6 − 1)² + (−1 − 1)² = 25 + 4 = 29.
(x − 1)² + (y − 1)² = 29
Problem 8 Inside, On or Outside
(a) (7 − 1)² + (0 + 2)² = 36 + 4 = 40: on the circle.
(b) (−2 − 1)² + (3 + 2)² = 9 + 25 = 34 is less than 40: inside.
(c) (5 − 1)² + (4 + 2)² = 16 + 36 = 52 is greater than 40: outside.
(a) on (b) inside (c) outside
Problem 9 Another Point on the Circle
Radius squared: (6 + 2)² + (−1 + 4)² = 64 + 9 = 73.
For (1, 4): (1 + 2)² + (4 + 4)² = 9 + 64 = 73.
Both points give r² = 73, so (1, 4) is on the circle.
Problem 10 Complete the Square: Basic
(x² − 6x + 9) + (y² + 10y + 25) = −18 + 9 + 25 = 16.
(x − 3)² + (y + 5)² = 16.
Center (3, −5), radius 4
Problem 11 Complete the Square: Irrational Radius
(x² + 8x + 16) + (y² − 2y + 1) = 4 + 16 + 1 = 21.
(x + 4)² + (y − 1)² = 21.
Center (−4, 1), radius √21 ≈ 4.58
Problem 12 Completing the Square with Large Terms
(x² − 12x + 36) + (y² − 4y + 4) = −20 + 36 + 4 = 20.
(x − 6)² + (y − 2)² = 20.
Center (6, 2), radius 2√5
Problem 13 Divide First
Divide by 2: x² + y² + 2x − 8y − 5 = 0.
(x² + 2x + 1) + (y² − 8y + 16) = 5 + 1 + 16 = 22.
(x + 1)² + (y − 4)² = 22.
Center (−1, 4), radius √22 ≈ 4.69
Problem 14 Circle, Point or Nothing?
(a) (x − 2)² + (y + 3)² = −13 + 4 + 9 = 0. Only (2, −3) satisfies it.
(b) (x − 2)² + (y + 3)² = −20 + 13 = −7. A sum of squares cannot be negative.
(a) the single point (2, −3) (b) no graph
Problem 15 Cell Tower Coverage
(a) (x + 3)² + (y − 5)² = 144.
(b) (6 + 3)² + (−3 − 5)² = 81 + 64 = 145, which is greater than 144, so the town is just outside.
(a) (x + 3)² + (y − 5)² = 144 (b) Not covered
Problem 16 Flower Bed in a Corner
The circle is tangent to both axes, so its center is 4 units from each: (4, 4).
(x − 4)² + (y − 4)² = 16
Problem 17 Circle Through Three Points
The angle at (0, 0) is 90°, so it is inscribed in a semicircle and the segment from (10, 0) to (0, 4) is a diameter.
Center = (5, 2). r² = 5² + 2² = 29.
Check (0, 0): 25 + 4 = 29.
(x − 5)² + (y − 2)² = 29
Problem 18 Ferris Wheel Heights
(a) x² + (y − 12)² = 100.
(b) 25 + (y − 12)² = 100, so (y − 12)² = 75 and y = 12 ± 5√3 ≈ 20.66 or 3.34.
(a) x² + (y − 12)² = 100 (b) ≈ 20.7 ft or ≈ 3.3 ft
Problem 19 Bike Path Through a Plaza
Substitute y = x + 1: (x − 1)² + (x + 1 − 2)² = 8, so 2(x − 1)² = 8 and (x − 1)² = 4.
x − 1 = ±2, so x = 3 or x = −1. Then y = 4 or y = 0.
(3, 4) and (−1, 0)
Problem 20 Tangent Line to a Circle
(a) (3 + 1)² + (4 − 2)² = 16 + 4 = 20. ✓
(b) The center is (−1, 2). The radius to (3, 4) has slope 4 − 23 + 1 = 12. The tangent is perpendicular, so its slope is −2.
y − 4 = −2(x − 3), so y = −2x + 10.
y = −2x + 10