Find equation of a circle with diameter endpoints calculator

William W. answered • 05/15/19

Math and science made easy - learn from a retired engineer

An equation for a circle fits the pattern (x - h)2 + (y - k)2 = r2 where (h, k) is the center of the circle and the radius is r.

The center of the circle would be at the midpoint of the diameter. The midpoint is determined by adding the two x values and dividing by 2 and adding the two y values and dividing by 2. So, for the x's: 3 + 10 = 13 and 13/2 = 6.5. For the y's: -4 + -12 = -16 and -16/2 = -8. So the circle center is at the point (6.5, -8).

The radius is half the diameter. To find the length of the diameter, use the distance formula which is:

d = sqrt[(y2 - y1)2 + (x2 - x1)2] = sqrt[(-12- -4)2 + (10 - 3)2] = sqrt(64 + 49) = sqrt(113). The radius is the half of that so sqrt(113)/2. The number in the circle equation is the radius squared so that would be [sqrt(113)/2]2. [sqrt(113)]2 is just 113 and 22 = 4 so the radius squared is 113/4.

The equation is then: (x - 6.5)2 + (y + 8)2 = 113/4

Use the midpoint formula to find center of the diameter and center of the circle, (h,k), then you use either set of coordinates with that information to solve for the radius

Mipoint is

x1 + x2/2 and y1 + y2/2

( (10+3)/2, (-12+(-4))/2 ) will give the coordinates of halfway along the diameter and the center of the circle

x = 13/2 =6.5

y = -16/2 = -8

(6.5, -8) are coordinates for half way along the diameter, and the center point of the circle (h, k)

To find r you can plug these values into the Equation for a circle

(x - h)2 + (y - k)2 = r2

Your circle equation so far is

(x - 6.5)2 + (y - (-8))2 = r2

Now just plug a set of the coordinates to solve for r

(10 - 6.5)2 + (-12 - (-8))2 =r2

(3.5)2 + (-4)2 = r2

12.25 + 16 = r2

28.25 = r2

√28.25 = r

5.32 =r

Finally, your circle equation is

(x - 6.5)2 + (y - (-8))2 = 28.25

You can check your coordinates for the radius directly with the classic distance formula as well, use either set of the given coordinates with the coordinates of the center

d = sqrt(6.5 - 3)2 + ((-8) - (-4))2

d = sqrt(3.5)2 + (-4)2

d = sqrt (12.25 + 16) = sqrt(28.25)= 5.3

I hope your find this useful if you have any questions just send me a message.

Equation of a Circle Calculator is a free online tool that displays the equation of a circle of a given input. BYJU’S online equation of a circle calculator tool makes the calculation faster, and it displays the equation in a fraction of seconds.

How to Use the Equation of a Circle Calculator?

The procedure to use the equation of a circle calculator is as follows:
Step 1: Enter the circle centre and radius in the respective input field
Step 2: Now click the button “Find Equation of Circle” to get the equation
Step 3: Finally, the equation of a circle of a given input will be displayed in the new window

What is the Equation of a Circle?

In geometry, a circle is a two-dimensional round shaped figure where all the points on the surface of the circle are equidistant from the centre point (c). The distance from the centre of the circle to the surface is called the radius (R). The equation of a circle can be calculated if the centre and the radius are known. Thus the equation of a circle is given by
(x-h)2 +(y-k)2 = r2
Where
(h, k) – centre coordinates
r – radius