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All courses Math II · G-GPE.1 41 of 73
Derive the equation of a circle from center/radius and complete the square to identify center/radius.

Write the equation of a circle from its center and radius

Problem
Write the center-radius equation of the circle with center (0, 0) and radius 5. Show how the center shifts enter the squared terms and how the radius determines the right side.
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Hint

Start with the standard form of a circle: (x-h)²+(y-k)²=r².

When the center is (0,0), the equation becomes x²+y²=r². Here r=5, so r²=25.

Solution walkthrough

01

Write center-radius form

\[(x-h)^2+(y-k)^2=r^2\]

A point (x,y) lies on a circle exactly when its distance from center (h,k) equals r; squaring the distance formula gives this standard form.

02

Source the center shifts

\[\text{center}~(h,k)=(0,0)~->~(x-0)^2+(y-0)^2=r^2\]

The center coordinates determine the numbers subtracted inside the squared terms. Both shifts are zero here.

03

Source and square the radius

\[r=5~->~r^2=25\]

The right side stores the square of the radius, not the radius itself.

04

Simplify and interpret

\[x^2+y^2=25\]

Removing the zero shifts gives the equation of all points whose distance from the origin is 5. For example, (5,0) satisfies 25+0=25.

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Another way

  1. Start from the distance equation sqrt(x squared+y squared)=5 and square both sides.

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Common mistake

Do not put 5 on the right side. Standard form uses r squared, so radius 5 gives 25.