InterMath | Workshop Support | Write Up Template
Title
Equation of a circle
Problem Statement
If a circle is centered at the origin, write an equation describing every point on the
circle. Your equation should use the radius of the circle and the x-coordinate and ycoordinate of ANY point on the circle.
Problem setup
I am trying to find an equation that can describe every point on a circle, where I use the
radius of a circle along with the x and y coordinates of any point on the circle.
Plans to Solve/Investigate the Problem
Prediction:
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Construct a circle in GSP with center of the circle located at the origin.
Plot a point on the circle.
Find any type of possible relationships, maybe right triangle?
Find an equation to represent the equation of any point on a circle, including x
and y coordinates and r, the radius of the circle
Investigation/Exploration of the Problem
1. First I plan to construct a circle in GSP (see figure below).
8
6
4
2
B
-5
5
10
15
-2
-4
-6
-8
2. I will construct the circle by using the origin (0, 0) as the center in my circle called
B. I constructed the center by constructing by center and radius. Then, I will place a
random point, A, on the circle and then connect point A and B to get my radius of the
circle.
4
A
2
B
-5
5
-2
-4
-6
3. After I created a random point on the circle and created line segment BA to represent
the radius of the circle, I made sure I used the snap point” option for both points A and B
to get an exact numerical value of the x and y coordinates so that is easy to work with.
4. Now I noticed that if I dropped a perpendicular line through point A, then I could form
a right triangle. This would allow me to take the value of x and the value of y to find r,
which is represented by line segment AB. Since the length of line segment BE is equal to
2cm and line segment AE is equal to 2, then x²+y²=r². Therefore, 2²+2²= r², which is
8=r². This means r=√8.
4
A
2
B
-5
E
5
-2
-4
A: (2.00, 2.00) xA = 2.00 yA = 2.00
B: (0.00, 0.00) xB = 0.00 yB = 0.00
In GSP, I then measure the length of line segment AB in centimeters, which ended up
being the same as above. This formula is an application of the Pythagoras theorem for
xA2+yA2 = 8.00
m BA 2 = 8.00 cm 2
right triangles.
By using the distance formula, I was able to find the distance ‘d’ between the points A (
x1, y1) and B (x2, y2).
xA-xB2+yA-yB2
= 2.83
m BA = 2.83 cm
m BE = 2.00 cm
m AE = 2.00 cm
This leads me to the conclusion that for any point on a circle, if you know the x and y
coordinates, then you can plug them into the formula x²+y²=r² to find the radius.
Extensions of the Problem
What if the circle is not centered at the origin?
Since the definition of a circle is a set of all points equidistant from a point. This
definition gives us terms of distances. Therefore, I can use the distance formula to come
up with an equation. The point I want to make my points equidistant from is called the
center and the distance is called the radius. So I am going to find the equation of the
circle with radius r and center (h,k). I am using (h,k) for the coordinates of the circle
rather than (x,y) because I have to reserve (x,y) for the points on the circle.
By the distance formula, the distance between (x,y) and (h,k) is given by this expression.
By the definition of a circle:
This is the standard equation of a circle with center (h,k) and radius r.
This means that if you were asked to write down the equation for a circle with a given
center and radius you should be able to do this by just putting the numbers into this
formula. You would substitute the x coordinate of the center for h and the y coordinate of
the center for k and the radius for r.
Author & Contact
Lauren Johnson, Middle Grades Cohort at Georgia College and State University
Lauren_johnson@ecats.gcsu.edu
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2