Math
/
Segment of a Circle

Segment of a Circle

Shiken premium Upgrade Banner

The Basics of Understanding Segments in a Circle

In geometry, a segment of a circle is defined as the area enclosed by a chord and the circumference, divided into minor and major segments. To work with these segments, it is essential to remember the formula for the area of a circle, which remains constant regardless of the angle being in radians or degrees.

Calculating Area in Radians

For finding the area of a segment in radians (marked in blue), one should know the central angle (x) and the radius. The formula for the minor segment is as follows:

Area = (1/2 * radius^2 * (x - sin(x)))

Rather than memorizing the two different formulas, it may be helpful to remember the major segment formula as follows:

Major segment area = Entire circle area - Minor segment area

To verify the calculation, add the areas of the minor and major segments, which should approximately equal the area of the whole circle. In simple words, minor segment + major segment = circle area (π * radius^2).

Calculating Area in Degrees

When the angle is given in degrees, the formula for the minor segment of a circle is:

Area = (1/2 * radius^2 * (x - sin(x)))

To find the area of the major segment in degrees, use the following formula:

Area = (1/2 * radius^2 * (x - sin(x)))

To calculate the major segment's area, subtract the minor segment's area from the entire circle's area.

Arc Lengths

The method for finding the arc length of a segment is identical to finding the sector's arc length. To calculate the arc length when the central angle (x) is in radians, use the formula:

Arc length = radius * x

For instance, if a circle C has a radius of 7 cm and a central angle of π/3, the arc length would be 7π/3 cm.

When the angle is in degrees, use the following formula:

Arc length = (π/180) * radius * x

For example, if Circle D has a radius of 5 cm and a central angle of 60 degrees, the arc length would be (π/180) * 5 * 60 = π/6 cm.

Key Takeaways

  • A segment of a circle is a portion of the circle enclosed by the circumference and a chord.
  • Segments can be minor or major, depending on their size.
  • Calculating the segment's area in a circle requires different formulas, depending on the angle at the center (in radians or degrees).
  • The principles for finding the arc length of a segment follow those of finding the sector's arc length.

Understanding the Concept of a Circle's Segment

To fully comprehend a segment of a circle, it is essential to understand its definition. A segment of a circle is the region between a chord and the circumference, divided into minor and major segments.

Calculating Segment Areas

The area of a segment of a circle can be determined by using specific formulas based on the angle at the center (in radians or degrees).

Area of a Segment

The area of a segment can be further divided into minor and major segments. When using the formula to determine the area, one is essentially calculating the minor segment's area.

Calculating the Area of a Segment in a Circle

To find the major segment's area in a circle, it is necessary to subtract the minor segment's area from the entire circle's area.

The formula for calculating the segment's area can be expressed in two ways. If the central angle (x) is given in radians, the formula is 1/2 × r ^ 2 × (x - sin(x)). On the other hand, if the angle is given in degrees, the formula is ((x × pi)/360 - sin(x/2)) × r ^ 2.

Both of these formulas offer a quick and efficient way to find the segment's area based on the given measurements. By plugging in the appropriate values for r (radius) and x (central angle), one can easily determine a segment's area in a circle.

Explore More Subject Explanations

Try Shiken Premium
for Free

14-day free trial. Cancel anytime.
Get Started
Join 20,000+ learners worldwide.
The first 14 days are on us
96% of learners report x2 faster learning
Free hands-on onboarding & support
Cancel Anytime