How to use the distance formula to find the radius of a circle?

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: ##sqrt((x_2-x_1)^2+(y_2-y_1)^2##

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If your two points are the center and a point on the circle…

Since the distance from any point on the circle to the center is considered the radius, plug in your given points into the distance formula.

Sample Question:
Find the radius of a circle, given that the center is at (2, –3) and the point (–1, –2) lies on the circle.

Work:
Using the distance formula, plug in points (2, -3) and (-1, -2) into the distance formula.

  1. ##sqrt((-2-(-3))^2+(-1-2)^2##
  2. ##=sqrt((1)^2+(-3)^2##
  3. ##=sqrt(1+9##
  4. ##=sqrt10##
  5. ##~~3.16##

Thus, the radius of the circle is ##sqrt10## which is approximately 3.16 rounded to two decimal places.

If your two points are two random points on the circle…

You can use the distance formula by plugging in the two points into the formula to find the diameter. However, remember to divide by two because you are looking for the radius.

Sample Question:
Find the radius of a circle, given that two random points are at (6, 3) and (10, 6)

Work:
Using the distance formula, plug in points (6, 3) and (10, 6) into the distance formula.

  1. ##sqrt((6-3)^2+(10-6)^2##
  2. ##=sqrt((3)^2+(4)^2##
  3. ##=sqrt(9+16##
  4. ##=sqrt25##
  5. ##=5##

However, the problem isn’t finished yet. The answer, ##5##, is the diameter of the circle. Since you are looking for the radius, you must divide ##5## by ##2##. (The radius is half of the diameter)

##5/2 = 2.5##

Thus, the radius of the circle is ##5/2## or ##2.5##



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