How To Find The Measure Of An Arc

Let's explore the fascinating world of circles and delve into understanding how to calculate the measure of an arc. Arcs, those curved segments of a circle's circumference, hold geometric significance, and knowing how to determine their measure is fundamental in various mathematical applications.

Understanding Arcs: A Foundation

Before diving into the methods of measuring arcs, it's crucial to grasp the basic definitions and types:

• Arc: A portion of the circumference of a circle.
• Minor Arc: An arc that is less than half of the circle.
• Major Arc: An arc that is more than half of the circle.
• Semicircle: An arc that is exactly half of the circle.

The measure of an arc is typically expressed in degrees, representing the central angle that subtends the arc. The central angle is an angle whose vertex is at the center of the circle and whose sides intersect the circle at the endpoints of the arc. The measure of a minor arc is equal to the measure of its central angle.

Methods to Find the Measure of an Arc

Several methods can be employed to determine the measure of an arc, depending on the available information. Let's explore these methods in detail:

1. Using the Central Angle

The most direct way to find the measure of an arc is when the measure of its central angle is known. As mentioned earlier, the measure of a minor arc is equal to the measure of its central angle.

Example:

If the central angle of an arc measures 60 degrees, then the measure of the arc is also 60 degrees.

For a major arc, the calculation is slightly different. Since a full circle is 360 degrees, the measure of a major arc is found by subtracting the measure of its corresponding minor arc (central angle) from 360 degrees.

Formula:

Measure of Major Arc = 360° - Measure of Central Angle

Example:

If the central angle of a minor arc measures 100 degrees, then the measure of the corresponding major arc is:

Measure of Major Arc = 360° - 100° = 260°

2. Utilizing Inscribed Angles

An inscribed angle is an angle formed by two chords in a circle that have a common endpoint. This common endpoint forms the vertex of the inscribed angle, and it lies on the circle's circumference. The measure of an inscribed angle is half the measure of its intercepted arc. The intercepted arc is the arc that lies in the interior of the inscribed angle and whose endpoints lie on the sides of the angle.

Theorem:

The measure of an inscribed angle is half the measure of its intercepted arc.

Formula:

Measure of Inscribed Angle = 1/2 * Measure of Intercepted Arc

To find the measure of the arc, we simply double the measure of the inscribed angle.

Formula:

Measure of Intercepted Arc = 2 * Measure of Inscribed Angle

Example:

If an inscribed angle measures 45 degrees, then the measure of its intercepted arc is:

Measure of Intercepted Arc = 2 * 45° = 90°

3. Employing Circumference and Arc Length

When the length of the arc and the circumference of the circle are known, we can determine the measure of the arc by using proportions.

Understanding the Relationship

The ratio of the arc length to the circumference is equal to the ratio of the arc's measure to 360 degrees (the total degrees in a circle).

Formula:

(Arc Length / Circumference) = (Measure of Arc / 360°)

To find the measure of the arc, we can rearrange the formula as follows:

Measure of Arc = (Arc Length / Circumference) * 360°

Example:

Suppose a circle has a circumference of 24 cm, and an arc on that circle has a length of 8 cm. To find the measure of the arc:

Measure of Arc = (8 cm / 24 cm) * 360° = (1/3) * 360° = 120°

Therefore, the measure of the arc is 120 degrees.

4. Working with Parallel Chords

When parallel chords intercept an arc, the arcs between those chords are congruent (have the same measure).

Theorem:

If two parallel lines intersect a circle, then the arcs between the parallel lines are congruent.

Application:

If you know the measure of one of the intercepted arcs, you automatically know the measure of the other arc between the parallel chords.

Example:

If two parallel chords intercept two arcs, and one of the arcs measures 70 degrees, then the other arc between the parallel chords also measures 70 degrees.

5. Using Tangents and Secants

The relationships between tangents, secants, and the arcs they intercept provide another avenue for finding arc measures.

• Tangent-Chord Angle: The angle formed by a tangent and a chord at the point of tangency. The measure of this angle is half the measure of the intercepted arc.
• Angle Formed by Two Tangents: The angle formed by two tangents drawn from an external point to a circle. The measure of this angle can be used to find the measure of the major and minor arcs intercepted by the tangents.
• Angle Formed by Two Secants: The angle formed by two secants that intersect outside the circle. The measure of this angle is half the difference of the measures of the intercepted arcs.
• Angle Formed by a Tangent and a Secant: The angle formed by a tangent and a secant that intersect outside the circle. The measure of this angle is half the difference of the measures of the intercepted arcs.

Formulas:

• Tangent-Chord Angle = 1/2 * Measure of Intercepted Arc
• Angle Formed by Two Tangents = 1/2 * (Measure of Major Arc - Measure of Minor Arc)
• Angle Formed by Two Secants = 1/2 * (Measure of Larger Arc - Measure of Smaller Arc)
• Angle Formed by a Tangent and a Secant = 1/2 * (Measure of Larger Arc - Measure of Smaller Arc)

Examples:

• Tangent-Chord Angle: If a tangent-chord angle measures 60 degrees, then the intercepted arc measures 120 degrees (2 * 60°).
• Angle Formed by Two Tangents: If the angle formed by two tangents is 40 degrees, and the minor arc measures x degrees, then the major arc measures (360 - x) degrees.
  • 40° = 1/2 * ((360 - x) - x)
  • 80° = 360 - 2x
  • 2x = 280
  • x = 140° (Minor Arc)
  • 360 - 140 = 220° (Major Arc)

6. Applying Vertical Angles and Linear Pairs

When dealing with intersecting lines (chords, secants, or tangents), remember the properties of vertical angles and linear pairs.

• Vertical Angles: Vertical angles are congruent (have the same measure). If intersecting lines create central angles that intercept arcs, the arcs intercepted by vertical angles are congruent.
• Linear Pairs: Linear pairs are supplementary (add up to 180 degrees). If a central angle forms a linear pair with another angle, you can find the measure of the central angle (and thus the arc) by subtracting the known angle from 180 degrees.

Example:

If two chords intersect inside a circle, forming vertical angles, and one of the intercepted arcs measures 85 degrees, then the arc intercepted by the vertical angle also measures 85 degrees.

Practical Applications

Understanding how to find the measure of an arc is not just a theoretical exercise; it has numerous practical applications in various fields:

• Engineering: Designing curved structures, bridges, and arches requires precise calculations involving arc lengths and measures.
• Architecture: Arches, domes, and curved facades are common architectural elements that rely on accurate arc measurements.
• Navigation: Calculating distances on maps and charts, especially when dealing with curved paths, involves understanding arc lengths on a sphere.
• Computer Graphics: Creating and manipulating curved shapes in computer graphics and animation relies on mathematical representations of arcs and curves.
• Manufacturing: Designing gears, pulleys, and other circular components requires precise calculations of arc lengths and angles.
• Astronomy: Understanding the orbits of celestial bodies, which are often elliptical, involves working with arcs and angles.

Common Mistakes to Avoid

• Confusing Arc Length and Arc Measure: Remember that arc length is the actual distance along the curve of the arc (measured in units like cm, inches, etc.), while arc measure is the angle that the arc subtends at the center of the circle (measured in degrees).
• Incorrectly Applying Inscribed Angle Theorem: Ensure that the angle you are using is indeed an inscribed angle (vertex on the circle) and that you are correctly identifying the intercepted arc.
• Forgetting to Consider Major Arcs: When dealing with central angles, remember to subtract from 360 degrees to find the measure of the corresponding major arc.
• Misusing Formulas for Tangents and Secants: Pay close attention to whether the angles are formed inside, outside, or on the circle when applying the tangent and secant theorems. Double-check which arcs are being intercepted.
• Assuming All Arcs Are Minor Arcs: Always consider whether you are dealing with a minor arc, major arc, or semicircle. The approach to calculation may differ.

Example Problems with Detailed Solutions

Here are some example problems that demonstrate the application of these methods:

Problem 1:

A circle has a central angle measuring 75 degrees. What is the measure of the arc intercepted by this angle? What is the measure of the corresponding major arc?

Solution:

• Since the central angle measures 75 degrees, the intercepted minor arc also measures 75 degrees.
• The measure of the corresponding major arc is 360° - 75° = 285°.

Problem 2:

An inscribed angle in a circle intercepts an arc. If the inscribed angle measures 32 degrees, what is the measure of the intercepted arc?

Solution:

• The measure of the intercepted arc is 2 * 32° = 64°.

Problem 3:

A circle has a circumference of 36 inches. An arc on the circle has a length of 9 inches. What is the measure of the arc?

Solution:

• Measure of Arc = (Arc Length / Circumference) * 360°
• Measure of Arc = (9 inches / 36 inches) * 360°
• Measure of Arc = (1/4) * 360° = 90°

Problem 4:

Two tangents are drawn to a circle from an external point. The angle formed by the two tangents is 50 degrees. What are the measures of the major and minor arcs intercepted by the tangents?

Solution:

Let the minor arc measure x degrees. Then the major arc measures (360 - x) degrees.

• 50° = 1/2 * ((360 - x) - x)
• 100° = 360 - 2x
• 2x = 260°
• x = 130° (Minor Arc)
• 360 - 130 = 230° (Major Arc)

Problem 5:

A secant and a tangent intersect outside a circle, forming an angle of 35 degrees. The larger intercepted arc measures 150 degrees. What is the measure of the smaller intercepted arc?

Solution:

Let the smaller arc measure y degrees.

• 35° = 1/2 * (150° - y)
• 70° = 150° - y
• y = 150° - 70°
• y = 80°

Conclusion

Finding the measure of an arc is a fundamental skill in geometry with widespread applications. Whether you're working with central angles, inscribed angles, arc lengths, or tangents and secants, understanding the relationships between angles and arcs is crucial. By mastering these methods and avoiding common mistakes, you'll be well-equipped to solve a wide range of problems involving arcs and circles. This knowledge not only strengthens your mathematical foundation but also opens doors to understanding and designing the curved world around us. Remember to practice regularly and apply these concepts in different contexts to solidify your understanding.