Basics & Circle Properties
Explore radius, diameter, circumference, and area relationships.
A Circle is a set of all points in a plane that are at a fixed distance (the Radius) from a central point. Fundamental measurements include:
- Diameter ($d$): The distance across the circle through the center ($d = 2r$).
- Circumference ($C$): The boundary perimeter length of the circle ($C = 2\pi r$).
- Area ($A$): The total surface region enclosed by the circle ($A = \pi r^2$).
Tangent to a Circle
Visualize how a tangent line meets the circle perimeter at exactly a right angle.
A Tangent is a straight line that touches the circle boundary at exactly one point. A core theorem of geometry states that:
The radius to the point of tangency is always perpendicular (90°) to the tangent line itself. This makes the tangent angle predictable and clean to compute.
Area of a Sector
Calculate the area of a circle slice and its arc length.
A Sector is a portion of a circle enclosed by two radii and an arc (resembling a pizza slice). Ratios rule this geometry:
- Arc Length ($s$): The curved boundary length of the slice ($s = \frac{\theta}{360^\circ} \cdot 2\pi r$).
- Sector Area ($A_{\text{sector}}$): The surface region of the slice ($A_{\text{sector}} = \frac{\theta}{360^\circ} \cdot \pi r^2$).
Segment of a Circle
Isolate the area bounded between a chord line and the circular arc.
A Segment is the region bounded by a chord and an arc. The calculation of the segment area is a subtraction of geometry parts:
A_segment = (θ / 360)·π·r² - 0.5·r²·sin(θ)