Diameter Formula
The formula for the diameter is d = 2r.
Arc Length Formula
The formula for arc length is X. C = (360e) - tangent, point of tangency, diameter, circumference.
Area Formula
The formula for the area of a circle is A = πr².
Properties of Circle Parts
- The measure of a central angle is the measure of the arc it contains.
- The measure of the inscribed angle is equal to half the measure of its intercepted arc.
- An inscribed angle is an angle with its vertex on the circumference of the circle with two sides that are chords.
- If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
- If an inscribed angle intercepts a diameter, then it is a right angle.
- If two inscribed angles intercept the same arc, then the angles are congruent.
- If two segments from the same external point are tangent to a circle, then they are congruent.
- A tangent line is perpendicular to a radius drawn to the point of tangency.
- If a polygon is circumscribed around a circle, then all of its sides are tangent.
Special Relationships
When chords, secants, and tangents intersect in a circle, on a circle, or outside of a circle, special relationships between the angle and arc measures are formed.
- Inside the Circle
- On the Circle
- Outside of the Circle -- 2 Secants
- Outside of the Circle -- 2 Tangents
- Outside of the Circle -- 1 Secant and 1 Tangent
Segment Lengths
- Intersecting Chords or Secants
- Intersecting Secants on the Exterior
- Intersecting Tangent and Secant on the Exterior
Equations of a Circle
- Standard Form: (x − h)² + (y − k)² = r²
- General Form: Ax² + By² + Cx + Dy + E = 0
Converting from General Form to Standard Form
- Given Equation
- Separate the x-related and y-related terms.
- Take half of the linear x term (cx) and square it. Add it to both sides of the equation.
- Take half of the linear y term (dy) and square it. Add it to both sides of the equation.
- Factor each polynomial.
For example, for the equation x² + y² + 8x - 2y - 8 = 0:
- x² + 8x + y² - 2y = 8
- x² + 8x + 16 + y² - 2y = 8 + 16
- x² + 8x + 16 + y² − 2y + 1 = 24 + 1
- (x + 4)² + (y − 1)² = 25
For more details and comprehensive information, you can download the Circles unit geometry reference sheet pdf free.
