Central Angle
A central angle is an angle formed by two radii in a circle that share the same endpoint at the center of the circle. The measure of the central angle is equal to the measure of the arc that it cuts out on the circumference of the circle.
Central angles are used in many geometric problems and play a crucial role in trigonometry, as the measure of a central angle is related to the lengths of the radii and the circumference of the circle. Central angles can be measured in degrees or radians.
A central angle can be found by dividing the length of the arc by the radius of the circle. If the length of the arc is known, the central angle can be calculated using the formula:
Central angle (?) = Arc length / Radius
On the other hand, if the measure of the central angle is known, the length of the arc can be calculated using the formula:
Arc length = Central angle (?) * Radius
It is important to note that central angles are not limited to right angles and can have any measure between 0° and 360°.
In a circle with 360°, the measure of the central angle is equal to the measure of the corresponding minor arc, but it is half the measure of the corresponding major arc. If the central angle cuts out a minor arc, the measure of the central angle is less than 180°, but if the central angle cuts out a major arc, the measure of the central angle is greater than 180°.
Central angles can also be used to find the area of sectors in a circle. A sector is a portion of a circle that is enclosed by two radii and an arc. The formula for the area of a sector is:
Area of sector = (Central angle / 360°) * ? * r^2
where ? is approximately equal to 3.14 and r is the radius of the circle.
Central angles play a crucial role in the study of circles and are important in geometry, trigonometry, and related areas of mathematics. Understanding central angles and their properties is essential for solving many geometric problems and for understanding the relationships between angles, arcs, and sectors in a circle.
Definitions:
- Arc: A portion of the circumference of a circle, defined by two points on the circumference.
- Radius: A line segment from the center of a circle to a point on the circumference of the circle.
- Degree: The unit of measurement for angles.
- Central Angle Measurement: The degree measurement of a central angle.
- Circumference: The distance around the edge of a circle.
Examples:
- Central Angle and Arc Length: Consider a circle with a radius of 5 units. If a central angle of 60 degrees is created by connecting two points on the circumference of the circle with two radii, the length of the arc that is cut off by the central angle can be determined. The length of the arc would be (60/360)2?*5 = 5?/3 units.
- Central Angle and Area of a Sector: The area of a sector can be found by determining the ratio of the central angle measurement to the total degrees in a circle (360) and multiplying the ratio by the area of the whole circle. For example, if the central angle of a circle with a radius of 7 units measures 120 degrees, the area of the sector would be (120/360)*?(7^2) = 14? square units.
- Central Angle and Length of a Chord: A chord is a line segment connecting two points on the circumference of a circle. The length of a chord can be determined by finding the length of the arc it spans and dividing by two. For example, if a central angle of 90 degrees cuts off an arc of length 20 units, the length of the chord would be 20/2 = 10 units.
- Central Angle and Inscribed Angle: An inscribed angle is an angle formed by two chords of a circle that share an endpoint on the circumference of the circle. The measure of an inscribed angle is half the measure of the central angle that cuts off the same arc as the inscribed angle. For example, if a central angle of 120 degrees cuts off an arc of length 40 units, the measure of the inscribed angle would be 120/2 = 60 degrees.
- Central Angle and Arc Length of a Major Arc: A major arc is an arc that is greater than 180 degrees. The length of a major arc can be found by subtracting the measure of the central angle from 360 and multiplying the result by the circumference of the circle. For example, if the central angle of a circle with a radius of 6 units measures 240 degrees, the length of the major arc would be (360-240)2?*6 = 480? units.
Quiz:
- What is a central angle?
- What is an arc?
- What is the relationship between the measure of a central angle and the length of the arc it cuts off on a circle?
- How can the area of a sector be determined?
- What is a chord and how can its length be found?
- What is an inscribed angle and how is its measure related to the measure of the central angle?
- What is a major arc and how is its length determined?
- What is a radius and how does it relate to a central angle?
- What is the unit of measurement for angles
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