Key takeaway: The equation of a circle with center (a, b) and radius R is (x − a)² + (y − b)² = R², which says every point on the circle lies exactly R units from the center.
Grade 10 – GeometryG-GPE.A.1Geometry and Measurement
Definition of Equation of a Circle
The equation of a circle is the distance formula in disguise. A point \((x, y)\) belongs to the circle when its distance from the center \((a, b)\) equals the radius \(R\). Squaring that statement removes the square root and leaves the standard form.
Reading it takes care with signs. The numbers subtracted inside the parentheses are the center’s coordinates, so \((x+3)\) means the first coordinate is \(-3\). The right side holds \(R^2\), so a square root is needed to recover the radius. A circle centered at the origin has the short form \(x^2 + y^2 = R^2\).
Formula and notation
The equation of a circle in standard form, for center \((a, b)\) and radius \(R\):
\[ (x – a)^2 + (y – b)^2 = R^2 \]
Multiplying out the squares gives the general form \(x^2 + y^2 + Dx + Ey + F = 0\), and completing the square returns to the standard form.
Key facts
- The center’s coordinates have the opposite signs of the numbers inside the parentheses.
- The constant on the right is the radius squared, not the radius.
- Completing the square converts the expanded version back to the readable one.
- A point lies on the circle exactly when its coordinates make the equation true.
Equation of a Circle example
A circular fishpond is drawn on a yard map with its center at \((-3, 5)\), and a lamp post at \((2, 11)\) stands exactly on its edge. Write the equation of a circle that models the pond.
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Horizontal gap: \(2-(-3) = 5\). Vertical gap: \(11-5 = 6\).
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Radius squared: \(R^2 = 5^2 + 6^2 = 25 + 36 = 61\).
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Put the center in: \((x – (-3))^2 + (y – 5)^2 = 61\).
The pond is modeled by \((x + 3)^2 + (y – 5)^2 = 61\), with radius \(\sqrt{61} \approx 7.8\).
Common mistake
Watch out
Hal reads the center of \((x + 3)^2 + (y – 5)^2 = 61\) as \((3, 5)\) and the radius as 61.
Why it is wrong
Two slips. The first coordinate is whatever is subtracted, and \(x + 3 = x – (-3)\), so the center is \((-3, 5)\). The 61 is \(R^2\), so the radius is \(\sqrt{61} \approx 7.8\), not 61.
Related terms and lessons
- Distance Formula
- Completing the Square
- Radius and Diameter
- Circumcenter
- Pythagorean Theorem
- Midpoint Formula
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