Central angle
A central angle is an angle formed by two radii of a circle, with its vertex at the center of the circle. The measure of a central angle equals the measure of the arc it intercepts. Since a full circle is 360 degrees, a central angle of 90 degrees cuts off a quarter-circle arc. This Grade 7 math skill from Saxon Math, Course 2 is foundational for constructing and reading circle graphs, dividing circles into sectors, and the arc length and sector area formulas that appear in high school geometry.
Key Concepts
Property A central angle is an angle that has its vertex at the center of a circle, and its sides are two radii.
Examples In a circle split into six equal sectors, each central angle measures $360^\circ \div 6 = 60^\circ$. A diameter creates a straight central angle of $180^\circ$, dividing the circle in half.
Explanation It is all in the name! The angle's corner is right at the circle's center. Its arms are two radii that cut out a pie slice shape called a sector, just like cutting a pizza.
Common Questions
What is a central angle?
A central angle is an angle whose vertex is at the center of a circle, formed by two radii. Its measure equals the arc it cuts off, measured in degrees out of the full 360-degree circle.
How does a central angle relate to circle sectors?
Each sector of a circle corresponds to a central angle. A sector with a 90-degree central angle is one quarter of the circle (90/360 = 1/4).
How do I find the measure of a central angle if I know the fraction of the circle?
Multiply the fraction by 360 degrees. A sector representing 1/3 of the circle has a central angle of (1/3) times 360 = 120 degrees.
What is the relationship between the central angle and the arc?
The arc measure in degrees equals the central angle measure. A 60-degree central angle intercepts a 60-degree arc — one-sixth of the full circle.
When do students learn about central angles?
Central angles are introduced in Grade 7 geometry. Saxon Math, Course 2 covers them in Chapter 9 alongside circle graphs and sector construction.
How do central angles help construct circle graphs?
To draw a circle graph sector representing 25% of data, calculate the central angle: 0.25 times 360 = 90 degrees. Use a protractor to draw a 90-degree central angle in the circle.
What are common mistakes with central angles?
Students sometimes confuse the central angle with the inscribed angle (vertex on the circle, not the center). A central angle equals the arc; an inscribed angle equals half the arc.