Circles
A circle is the collection of points equidistant from a given point, called the center. A circle is named after its center point. The distance from the center to any point on the circle is called the radius, (r), the most important measurement in a circle. If you know a circle’s radius, you can figure out all its other characteristics. The diameter (d) of a circle is twice as long as the radius (d = 2r) and stretches between endpoints on the circle, passing through the center. A chord also extends from endpoint to endpoint on the circle, but it does not necessarily pass through the center. In the figure below, point C is the center of the circle, r is the radius, and AB is a chord.
Tangent Lines
Tangents are lines that intersect a circle at only one point. Here’s the first rule about them: A radius whose endpoint is the intersection point of the tangent line and the circle is always perpendicular to the tangent line. See for yourself:
And the second rule: Every point in space outside the circle can extend exactly two tangent lines to the circle. The distance from the origin of the two tangents to the points of tangency are always equal. In the figure below, XY = XZ.
Tangents and Triangles
Tangent lines are most likely to appear with triangles.
Using the figure below find the area of the triangle.
In this question since QR and RS are perpendicular, and angle RQS is 60°, triangle QRSis a 30-60-90 triangle. The image tells you that side QR, the side opposite the 30° angle equals 4. Side QR is the height of the triangle. To calculate the area, you just have to figure out which of the other two sides is the base. Since the height and base of the triangle must be perpendicular to each other, side RS must be the base. To find RS, use the 1:
:2 ratio (Math Sat - Triangles). RS is the side opposite 60°, so it’s the 
side: RS = 4
. The area of triangle QRS is 1/2(4)(4
) = 8
. Central Angles and Inscribed Angles
An angle whose vertex is the center of the circle is called a central angle.
The degree of the circle cut by a central angle is equal to the measure of the angle. If a central angle is 25º, then it cuts a 25º arc in the circle.
An inscribed angle is an angle formed by two chords originating from a single point.
An inscribed angle will always cut out an arc in the circle that is twice the size of the degree of the inscribed angle. If an inscribed angle has a degree of 40, it will cut an arc of80º in the circle.
If an inscribed angle and a central angle cut out the same arc in a circle, the central angle will be twice as large as the inscribed angle.
Circumference of a Circle
The circumference is the perimeter of the circle. The formula for circumference of a circle is
where r is the radius. The formula can also be written C = πd, where d is the diameter.
Arc Length
An arc is a part of a circle’s circumference. An arc contains two endpoints and all the points on the circle between the endpoints. By picking any two points on a circle, two arcs are created: a major arc, which is by definition the longer arc, and a minor arc, the shorter one.
Since the degree of an arc is defined by the central or inscribed angle that intercepts the arc’s endpoints, you can calculate the arc length as long as you know the circle’s radius and the measure of either the central or inscribed angle.
The arc length formula is
where n is the measure of the degree of the arc, and r is the radius.
Area of a Circle
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