Circles and Shapes

Answers

1) (2, 0) and 4 2) (-4, 5) and 5 3) (x - 4)^2 + (y - 3)^2 = 25 or (x - 4)^2 + (y + 3)^2 = 25 4) (-1, 3) and 5

Graphing Circles

A circle centered at the origin of a set of Cartesian coordinates gives the length of the radius as r^2 = x^2 + y^2. This also can be written as r^2 = (x - 0)^2 + (y - 0)^2.

Radius of the Circle

A line drawn from the Center of the Circle to any edge. Two radii connected as a straight line make a diameter.

Circumference of a Circle

Equation is Pi * diameter or 2 * Pi * radius.

Area of a Circle

Equation is Pi * radius squared or .25 * Pi * diameter squared.

Problem 4

Find the coordinates of the center and length of the radius for the following equation of a circle: x^2 + y^2 - 6y + 2x = 15

Problem 2

Give the coordinates of the center and length of the radius given the equation (x + 4)^2 + (y - 5)^2 = 25.

Problem 1

Give the coordinates of the center and length of the radius given the equation (x - 2)^2 + y^2 = 16.

Problem 3

Given the point on the circumference (0, 0), radius of 5, and partial equation (x - 4)^2, write the full equation.

Moving Circles from the Origin

Moving circles from the origin in a positive x or positive y direction, use the format variable minus new center coordinate. Moving them in a negative -x or -y direction, use the format variable plus new center coordinate.

Auxiliary Information - Pi

Pi is the constant relationship between the diameter of a circle and its circumference. This is 22/7 or 3.14 with no units.

Circumference of the Circle

The outer edge of the circle is called the circumference. This is the same thing as the perimeter of rectangles and triangles.

Completing the Square

The x- and y-equations, when expanded, are quadratic equations. Given a quadratic equation, it can be put into circle format by completing the square. Easiest way to do this, when the coefficient in front of the square term is 1, is to divided the first power coefficient by 2, and that becomes the off-set from the variable.

Circles

Three parts of the circle are: 1) the center; 2) the radius; and, 3) the circumference.

Cone is a 3D spinning triangle

Volume is 1/3 * Pi * radius squared times height or 1/12 * Pi * diameter squared times height.

Sphere is a 3D circle

Volume is 4/3 * Pi * radius cubed or 1/6 * Pi * diameter cubed.

Pythagorean Theorem

When a radius is drawn in a circle that is on a Cartesian coordinate system, and a horizontal radius is drawn, a vertical is dropped from the intersection of the first radius and the circumference to make a triangle.

Center of the Circle

When two or more lines are drawn from one edge of the circle to the other edge, the lines cross at the center.