Equation of a Circle Notes and Worksheets - Lindsay Bowden - Free Printable

Let's go through the Equation of a Circle worksheet step by step and solve all the problems. This is a great review of the standard form of the equation of a circle:

$$
(x - h)^2 + (y - k)^2 = r^2
$$

Where:
- $(h, k)$ is the center of the circle,
- $r$ is the radius,
- $(x, y)$ is any point on the circle.

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🔹 Fill in the blanks at the top:

Given:
$$
(x - h)^2 + (y - k)^2 = r^2
$$

- $(h, k) =$ Center of the circle
- $(x, y) =$ Any point on the circle
- $r =$ Radius

> ✔ The equation of a circle is used to graph a circle or find parts of the circle such as the center, radius, or a point on the circle.

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## ✔ Part 1: Find the center and radius of each circle

We use the standard form: $(x - h)^2 + (y - k)^2 = r^2$

1. $x^2 + y^2 = 25$

This is equivalent to: $(x - 0)^2 + (y - 0)^2 = 5^2$

- Center: $(0, 0)$
- Radius: $5$

✔ Answer: Center: (0, 0), Radius: 5

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2. $(x + 8)^2 + (y - 3)^2 = 100$

Rewrite: $(x - (-8))^2 + (y - 3)^2 = 10^2$

- Center: $(-8, 3)$
- Radius: $10$

✔ Answer: Center: (-8, 3), Radius: 10

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3. $(x - 4)^2 + y^2 = 1$

Rewrite: $(x - 4)^2 + (y - 0)^2 = 1^2$

- Center: $(4, 0)$
- Radius: $1$

✔ Answer: Center: (4, 0), Radius: 1

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4. $(x - 5)^2 + (y + 1)^2 = 144$

Rewrite: $(x - 5)^2 + (y - (-1))^2 = 12^2$

- Center: $(5, -1)$
- Radius: $12$

✔ Answer: Center: (5, -1), Radius: 12

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5. $x^2 + (y + 2)^2 = 18$

Rewrite: $(x - 0)^2 + (y - (-2))^2 = (\sqrt{18})^2$

- Center: $(0, -2)$
- Radius: $\sqrt{18} = 3\sqrt{2}$

✔ Answer: Center: (0, -2), Radius: $3\sqrt{2}$

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6. $x^2 + y^2 = 16$

This is: $(x - 0)^2 + (y - 0)^2 = 4^2$

- Center: $(0, 0)$
- Radius: $4$

✔ Answer: Center: (0, 0), Radius: 4

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## ✔ Part 2: Write the equation of the circle with given center and radius

Use: $(x - h)^2 + (y - k)^2 = r^2$

1. Center: $(0, 0)$, Radius: 4

$$
(x - 0)^2 + (y - 0)^2 = 4^2 \Rightarrow x^2 + y^2 = 16
$$

✔ Answer: $x^2 + y^2 = 16$

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2. Center: $(2, -3)$, Radius: 15

$$
(x - 2)^2 + (y + 3)^2 = 15^2 \Rightarrow (x - 2)^2 + (y + 3)^2 = 225
$$

✔ Answer: $(x - 2)^2 + (y + 3)^2 = 225$

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3. Center: $(-7, 0)$, Radius: 10

$$
(x + 7)^2 + (y - 0)^2 = 10^2 \Rightarrow (x + 7)^2 + y^2 = 100
$$

✔ Answer: $(x + 7)^2 + y^2 = 100$

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4. Center: $(-5, -1)$, Radius: 2

$$
(x + 5)^2 + (y + 1)^2 = 2^2 \Rightarrow (x + 5)^2 + (y + 1)^2 = 4
$$

✔ Answer: $(x + 5)^2 + (y + 1)^2 = 4$

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5. Center: $(0, 0)$, Radius: 1

$$
x^2 + y^2 = 1
$$

✔ Answer: $x^2 + y^2 = 1$

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6. Center: $(4, 8)$, Radius: 22

$$
(x - 4)^2 + (y - 8)^2 = 22^2 \Rightarrow (x - 4)^2 + (y - 8)^2 = 484
$$

✔ Answer: $(x - 4)^2 + (y - 8)^2 = 484$

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## ✔ Part 3: Graph the circle

Example: $(x - 1)^2 + (y - 1)^2 = 9$

- Center: $(1, 1)$
- Radius: $\sqrt{9} = 3$

To graph:
- Plot center at $(1, 1)$
- Draw a circle with radius 3 units in all directions.

(You would draw this on the grid provided.)

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## ✔ Part 4: Use the graph to write the equation of the circle

Look at the right-hand graph:

- The center appears to be at $(-2, 1)$
- The radius: from center to edge is 3 units (counting grid squares)

So:
- $h = -2$, $k = 1$, $r = 3$
- Equation: $(x + 2)^2 + (y - 1)^2 = 9$

✔ Answer: $(x + 2)^2 + (y - 1)^2 = 9$

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## ✔ Final Summary Answers

Fill in the Blanks:

- $(h, k) =$ Center
- $(x, y) =$ Point on the circle
- $r =$ Radius
- The equation is used to graph a circle or find the center, radius, or a point on the circle.

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Part 1: Find center and radius

| Equation | Center | Radius |
|--------|--------|--------|
| 1. $x^2 + y^2 = 25$ | (0, 0) | 5 |
| 2. $(x+8)^2 + (y-3)^2 = 100$ | (-8, 3) | 10 |
| 3. $(x-4)^2 + y^2 = 1$ | (4, 0) | 1 |
| 4. $(x-5)^2 + (y+1)^2 = 144$ | (5, -1) | 12 |
| 5. $x^2 + (y+2)^2 = 18$ | (0, -2) | $3\sqrt{2}$ |
| 6. $x^2 + y^2 = 16$ | (0, 0) | 4 |

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Part 2: Write equations

| Center | Radius | Equation |
|-------|--------|----------|
| (0, 0) | 4 | $x^2 + y^2 = 16$ |
| (2, -3) | 15 | $(x - 2)^2 + (y + 3)^2 = 225$ |
| (-7, 0) | 10 | $(x + 7)^2 + y^2 = 100$ |
| (-5, -1) | 2 | $(x + 5)^2 + (y + 1)^2 = 4$ |
| (0, 0) | 1 | $x^2 + y^2 = 1$ |
| (4, 8) | 22 | $(x - 4)^2 + (y - 8)^2 = 484$ |

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Part 3 & 4:

- Graph: Center at (1,1), radius 3 → draw circle accordingly.
- From graph: Equation is $(x + 2)^2 + (y - 1)^2 = 9$

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✔ All problems solved! Let me know if you'd like this formatted into a printable answer sheet.