Derive the Equation of a Circle

Lesson 1 of 2: Building Standard Form

In this lesson:

• Derive standard form using the Pythagorean Theorem
• Read center and radius from any standard form equation
• Verify whether a point lies on a circle

Learning Objectives

After this lesson, you should be able to:

1. Derive the equation of a circle using the Pythagorean Theorem
2. Write circle equations in standard form:
3. Read center and radius from a standard form equation
4. Verify whether a given point lies on a circle
5. Explain why standard form captures all points at distance from the center

What Is a Circle, Exactly?

You already know the definition:

A circle is the set of all points at a fixed distance (the radius) from a center point

Every point on the circle is exactly that distance away — no closer, no farther.

How do we express this idea algebraically?

A Circle in the Coordinate Plane

A point lies on the circle if and only if its distance from equals .

The Right Triangle Inside Every Circle

Horizontal leg ; vertical leg ; hypotenuse (the radius)

The Standard Form Equation

Apply the Pythagorean Theorem to the right triangle inside the circle:

• — square of the horizontal leg
• — square of the vertical leg
• — square of the hypotenuse (radius)

Distance Formula: Same Result

Squaring both sides eliminates the radical:

The distance formula and Pythagorean Theorem produce the same equation ✓

Standard Form: What Each Term Means

Each term is the Pythagorean Theorem applied to the right triangle inside the circle.

Worked: Write the Equation and Verify

Given: Center , radius .

Verify: Is on this circle?

Quick Check

Write the equation of a circle with center and radius .

Write it before the next slide...

Quick Check — Answer

• → first term simplifies to
• →
• — the right side stores , not

From Building to Reading

So far: given a center and radius, we have built the equation.

Now: given an equation, we extract the center and radius.

Template:

• Center: — the values subtracted from and
• Radius:

Reading Standard Form

The values subtracted from and give the center; gives the radius.

Example A: Read the Equation

• Center: , → center is
• Radius: →

means , not — always take the square root.

Example B: The Sign Trap

Rewrite: , so .

• Center: — not !
• Radius:

means , not

Example C: Centered at the Origin

Rewrite:

• Center: — the origin
• Radius:

The simplest case: , so both subtracted values vanish.

Your Turn: Write the Equation

Given: Center , radius

Hint:

Write the standard form equation — leave it in exact form.

Try before the next slide...

Your Turn — Answer

• : first term is , not
• : second term is
• : radius is , right side is

Verifying Points on a Circle

To check if lies on a circle, substitute the coordinates and evaluate:

• If left side : the point lies on the circle ✓
• If left side : the point is inside the circle
• If left side : the point is outside the circle

The equation is an exact algebraic test for circle membership.

Worked: Is (5, 1) on the Circle?

The right side of is also .

Yes — lies on the circle at distance from center ✓

Quick Check

Is on the circle ?

Substitute and compare to 8...

✓

Yes — is on the circle at distance from center .

Practice

1. Find the center and radius of

2. Write the equation: center , radius

3. Is the point on the circle ?

Work through all three, then advance for answers.

Practice — Answers

1. : center , radius
   (Note: , so )

2. Center , radius :

3. ✓ — is on

Key Takeaways

✓ — center , radius
✓ Derived from the Pythagorean Theorem — not arbitrary algebra
✓ Radius — the equation stores , not
✓ Verify a point: substitute and check if the left side equals

means — always rewrite to see the sign
Right side — square-root it to find

Coming Up: Lesson 2

This is a circle equation — but where is the center? Where is the radius?

Lesson 2: Use completing the square to convert any circle equation to standard form.