1. Degrees and Radians
2. Circle Central Angle
3. Quadrant and Semicircle
4. Angles and Chords
There are a variety of situations where specific angles in a circle can appear.
It’s quite useful to be aware of some common properties of these angles seeing as the circle is one of the more common shapes to encounter in Geometry.
Degrees and Radians in a Circle
When it comes to degree measure, there is 360° in a full size circle.
But along with degrees, there is also radian measure, for which there are 2π radians in a full size circle.
The angles that occur in a circle can be measured with both degree and radian measure.
This page will show examples of angles which are measured with degrees.
More information on radian measure can be seen here.
Central in a Circle
The central angle of a circle is an angle created by 2 radius lines, joined together by a vertex at the center of the circle.
The circle below has a central angle 70° in size.
Circle Semicircle and Quadrant
With a standard circle, a quadrant is the size of one quarter or one fourth of the whole circle.
The angle that is present inside a quadrant is 90°, which means that the angle outside the quadrant inside the circle is 270°.
A semicircle half of a circle, and is thus the size of 2 quadrants combined together. So the size of the angle inside a semicircle is 180°, as is the angle outside of the semicircle.
Triangles in a Semicircle
If you have a triangle inside a semicircle, it will always be a right angle triangle, with the angle NOT alongside the diameter being 90°.
Angles in a Circle Rules
Chords and Inscribed Angle
An angle that is inscribed by 2 chords inside a circle, will have a vertex that is on the circumference of a circle.
The size of the inscribed angle, is half of the size of the arc that is formed by the other ends of the 2 chord lines.
Angles between Intersecting Chords
It turns out that four angles are formed when 2 chord lines inside a circle intersect each other.
The four angles are in fact 2 pairs of angles that happen to be vertically opposite to each other.
Which means:∠1 = ∠4 AND ∠2 = ∠3
We can observe this with actual angle measurements in the image below.
