Circles
A circle is a fundamental geometric shape defined as the set of all points in a plane that are equidistant from a fixed center point. The distance from the center to any point on the circle is called the radius. If you connect any two points on the circle with a straight line passing through the center, you get a diameter. The diameter is twice the length of the radius.
Key Concepts:
Center: The fixed point in the middle of the circle from which all points on the circle are equidistant.
Radius: The distance from the center of the circle to any point on the circle.
Diameter: A straight line passing through the center of the circle and connecting two points on the circle. The diameter is twice the length of the radius.
Circumference: The perimeter or boundary of the circle. The formula for the circumference (C) is C=2πr, where r is the radius.
Pi (π): An irrational number approximately equal to 3.14159. It is used in formulas involving circles, such as the circumference and area.
Arc: A portion of the circumference of the circle. A minor arc is smaller than a semicircle, while a major arc is larger.
Sector: The region enclosed by two radii and the corresponding arc.
Chord: A straight line segment whose endpoints are on the circle.
Segment of a Circle:
A segment in a circle is the region enclosed by a chord (a straight line connecting two points on the circle) and the corresponding arc (the curved part of the circle between those two points).
There are two types of segments:
Minor Segment: The smaller region of the circle formed by a chord and the arc.
Major Segment: The larger region of the circle formed by the chord and the arc.
For example, if you draw a line across a pizza, the slice you get between that line (the chord) and the edge of the pizza (the arc) is the segment.
Tangent to a Circle:
A tangent is a straight line that touches a circle at exactly one point. This point is called the point of tangency.
A key property of a tangent line is that it is perpendicular to the radius of the circle drawn to the point of tangency. This means that the angle between the tangent line and the radius at the point of tangency is 90 degrees.
For instance, imagine a circle and a straight line that just barely touches it at one single point; this line is the tangent to the circle.
Key Differences:
A segment is an area within the circle enclosed by a chord and its corresponding arc.
A tangent is a line outside the circle that touches it at exactly one point.
Circumference of a Circle
The circumference is the distance around the edge of a circle. It’s similar to the perimeter of a polygon but specific to circles.
To find the circumference (C) of a circle, you multiply the radius (r) by 2π. This formula is used to calculate the distance around the edge or perimeter of a circle.
Area of a Circle
The area of a circle is the amount of space inside the circle. It tells you how many square units fit inside the circle.
To find the area of a circle, you square the radius (r) and multiply it by π. This formula applies to circles of any size.
Exercises (Circumference)
Complete the following exercises on your own, then check your answers by watching the video below.
Exercises (Area)
Complete the following exercises on your own, then check your answers by watching the video below.
Watch to See (C) Answers
Watch to See Area Answers
Circle Theorems
Circle theorems explore the intriguing relationships between the angles, chords, tangents, and other elements associated with a circle. Understanding these theorems allows us to easily solve complex problems and provides insight into the symmetrical and harmonious nature of circles.
