| For a circle drawn in the plane, if
r denotes the radius and
the center of the circle is at (h,k) then the equation of a circle
in standard form (based on the distance formula) is:
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| 1.4 Circles |
| A circle is the collection of points in the plane that are a fixed distance from a given point. The fixed distance is called the radius of the circle and the given point the center of the circle. |
| For a circle drawn in the plane, if
r denotes the radius and
the center of the circle is at (h,k) then the equation of a circle
in standard form (based on the distance formula) is:
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Move the sliders to see how changes in the radius r and center, (h,k), affect the size and placement of the circle. Also, as you vary the parameters, note the equation of the new circle. |
In the following exercise, you are given a quadratic equation in x and y. First complete the square in order to find the equation in standard form. Then use this form to find the center of the circle and the radius of the circle. When you are done with your calculations, click "Graph it" in order to check your answers and see the graph of the circle.
To begin, or to get a new equation, click on "New".
| Completing the square to find center and radius |
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A nice geometric problem that uses both circles and lines is sometimes called the Broken Wheel Problem. Give it a try.
| Sometimes we are not interested in the
full circle but only in a semicircle.
Let's find equations of 4 semicircles cut out of the circle on
the right which has equation |
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| In order to find the equation for the
upper
semicircle: Solve the equation for (y + 1)2.
Then take the positive square root of both sides and solve for y
itself.
For the lower semicircle: Solve the equation for (y + 1)2. Then take the negative square root of both sides and solve for y itself. For the equation for the left semicircle: Solve for (x - 2)2. Then take the negative square root of both sides and solve for x. For the right semicircle: Solve for (x - 2)2. Then take the positive square root of both sides and solve for x. |
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This exercise requires you to find the formula for a given semicircle.
Move the pointer over the "?" in the typing area of you would like help
in typing mathematics. Then click in the typing area (again) to begin typing.
| Finding equations of semicircles |
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