St. James 03-31-2025 Circle Equation Review

Draw

Write your solutions to BOTH puzzles. Be prepared to explain your reasoning.

​r²

​9

​Applications of Equations of a Circle

​1. Architecture & Engineering

• Application: Designing circular structures like domes, tunnels, and arches.

• Suggested Image: The Pantheon dome in Rome or a modern circular bridge.

2. Physics (Orbital Motion)

• Application: Describing planetary orbits, motion of satellites, and circular motion in physics.

• Suggested Image: Diagram of Earth’s orbit around the Sun or a satellite orbiting Earth.

3. Computer Graphics & Game Design

• Application: Used in animation, collision detection, and 3D rendering.

• Suggested Image: A game development interface showing collision boundaries using circles.

• Suggested Image: A CT scan image showing circular cross-sections of the human body.

​Applications of Equations of a Circle

4. Mechanical Engineering (Gears & Wheels)

• Application: Designing gears, wheels, and mechanical components that rely on circular motion.

• Suggested Image: A close-up of gears in a clock or a car wheel.

5. Sports & Recreation

• Application: Calculating the trajectory of a ball in basketball, soccer, or golf.

• Suggested Image: A basketball going through a hoop, with the trajectory modeled as a circular path.

6. Medicine & Biology (Medical Imaging)

• Application: Used in CT scans and MRI technology to analyze cross-sectional images.

• Suggested Image: A CT scan image showing circular cross-sections of the human body.

Real Uses of Circle Equations

​9

Why & When to Use Each Form

Why & When to Use Each Form

Standard to General Form

WRITE the STEPS - not a screenshot!

Any time you see: x² and x or y² and y, , you have to complete the square.

What if it's not in standard form?

2. Imagine what constant makes it a "perfect trinomial" (ADD to other side to balance!)

1. Put each variable in order

(I don't do this step on the left)
3. Combine all constants on right
4. Write in standard form!

Dropdown

Equation of Circle

The equation of circle provides an ​

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to describe a circle, given the ​ ​

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and the length of the ​

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of a circle. The equation of a circle is different from the formulas that are used to calculate the area or the circumference of a circle. This equation is used across many problems of circles in ​

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geometry.​

Dropdown

What is the Equation of Circle?

An equation of a circle represents ​

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of a circle in a Cartesian plane. If we know the coordinates of the ​

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and the length of its​

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, we can write the equation of a circle. The equation of circle represents​

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of the circle.

Dropdown

A circle represents the ​

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whose distance from a ​

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is a constant value. This fixed point is called the center of the circle and the constant value is the​

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of the circle. The standard equation of a circle with center at ​

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and radius r is ( x - x₁)² + ( y - y₁)² = r² .

Multiple Choice

What is the standard equation of a circle?

• (x-k)² + (y-h)² = r²

• (x+h)² + (y+k)² = r²

• (x-h)² + (y-k)² = r²

• (k-x)² + (h-y)² = r²

Math Response

Determine the equation of the circle.

Type answer here

Correct Answer: \left(x-2\right)^2+\left(y-7\right)^2=4

Deg^°

Rad

Math Response

Type answer here

Correct Answer: \left(x-5\right)^2+\left(y+2\right)^2=16

Deg^°

Rad

Math Response

Type answer here

Correct Answer: \left(x-5\right)^2+\left(y-6\right)^2=16

Deg^°

Rad

Math Response

Type answer here

Correct Answer: \left(x-6\right)^2+\left(y+4\right)^2=10

Deg^°

Rad

Math Response

Type answer here

Correct Answer: \left(x+4\right)^2+\left(y-6\right)^2=13

Deg^°

Rad

Math Response

Type answer here

Correct Answer: \left(x+3\right)^2+\left(y+4\right)^2=32

Deg^°

Rad

Math Response

Determine the equation of the circle with:

center ( 2, 3 ) containing the point ( 7, 15 ).

Type answer here

Correct Answer: \left(x-2\right)^2+\left(y-3\right)^2=169

Deg^°

Rad

Math Response

Determine the equation of the circle with:

radius 9 and center (−1,−4).

Type answer here

Correct Answer: \left(x+1\right)^2+\left(y+4\right)^2=81

Deg^°

Rad

Math Response

Determine the equation of the circle with :

center (−6 , −2 ) containing the point ( 3, −6 ).

Type answer here

Correct Answer: \left(x+6\right)^2+\left(y+2\right)^2=97

Deg^°

Rad

Multiple Choice

The center and radius of the given circle $$\left(x-3\right)^2+\left(y+8\right)^2=39$$

• $$\left(3,-8\right),\ r=39$$

• $$\left(-3,-8\right),\ r=\sqrt[]{39}$$

• $$\left(-3,8\right),\ r=\sqrt[]{39}$$

• $$\left(3,-8\right),\ r=\sqrt[]{39}$$

Multiple Choice

Which equation of a circle will have a graph that lies entirely in the first quadrant?

• $$\left(x-4\right)^2+\left(y-5\right)^2=9$$

• $$\left(x+4\right)^2+\left(y+5\right)^2=9$$

• $$\left(x+4\right)^2+\left(y+5\right)^2=25$$

• $$\left(x-5\right)^2+\left(y-4\right)^2=25$$

Multiple Choice

Select the graph of the circle: $$\left(x-1\right)^2+\left(y+2\right)^2=25$$  

Multiple Choice

Select the graph of the circle: $$\left(x-3\right)^2+y^2=4$$  

Multiple Choice

What is the equation of the circle?

• $$\left(x-4\right)^2+\left(y+2\right)^2=3$$

• $$\left(x+4\right)^2+\left(y-2\right)^2=3$$

• $$\left(x-4\right)^2+\left(y+2\right)^2=9$$

• $$\left(x+4\right)^2+\left(y-2\right)^2=9$$

Multiple Choice

Which graph corresponds to

(x-4)²+ (y+4)²=25

• A

• B

• C

• D

Multiple Choice

Which graph corresponds to

(x+2)²+ (y-3)²=36

• A

• B

• C

• D

Multiple Choice

See Picture

• A

• B

• C

Multiple Choice

In the equation (x+2)²+(y-3)²=4, the radius of the circle is...

• 4
• 2
• 8
• 16

Multiple Choice

In the equation (x-3)²+(y-2)²=16, the center of the circle is...

• (3,2)
• (-3, -2)
• (-2, -3)
• (2, 3)

Multiple Choice

In the equation (x+2)²+(y-3)²=4, the center of the circle is at...

• (-2, 3)
• (-3, 2)
• (2, -3)
• (3, -2)

Multiple Choice

In the equation (x+2)²+(y+3)²=49, the radius of the circle is...

• 24.5
• 98
• 7
• 49

Multiple Choice

Which of the following equations of circle in general form given the center at (5, -8) and radius of 11 units?

• x² + y² -10x + 16y + 32 = 0

• x² + y² -10x + 16y - 32 = 0

• (x - 5)² + (y + 8)² = 11

• (x - 5)² + (y + 8)² = 121

Multiple Choice

Determine the equation of the circle in standard form: x² + y² + 6x - 12y + 20 = 0

• (x-3)² + (y+6)² = 25
• (x+3)² + (y-6)² = 25
• (x-3)² + (y+6)² = 45
• (x+3)² + (y-6)² = 5

Multiple Choice

Determine the equation of the circle in standard form:
x² + y² + 4x + 6y - 36 = 0

• (x+2)² + (y+3)² = 49
• (x+2)² + (y+3)² = 13
• (x-2)² + (y-3)² = 49
• (x-2)² + (y-3)² = 13

Multiple Choice

Determine the equation of a circle with center (5,-3) and radius of 9

• x²+y²-10x+6y-47=4
• x²+y²-10x-6y-47=0
• 2x²+y²+10x+6y-47=0
• x²+y²-10x+6y-47=0

Multiple Choice

Determine the equation of the circle in standard form:
x² + y² + 4x + 6y - 36 = 0

• (x+2)² + (y+3)² = 49
• (x+2)² + (y+3)² = 13
• (x-2)² + (y-3)² = 49
• (x-2)² + (y-3)² = 13

Multiple Choice

Determine the equation of a circle with center (5,-3) and radius of 9

• x²+y²-10x+6y-47=4
• x²+y²-10x-6y-47=0
• 2x²+y²+10x+6y-47=0
• x²+y²-10x+6y-47=0

Poll

What is your level of confidence of today's lesson?