Area Of A Segment Of A Circle, How Do I Find The Area Of A Segment?

The area of a segment of a circle is a fundamental concept in geometry, used to calculate the area of the region bounded by a chord and an arc. To find the area of a segment, we need to know the radius of the circle and the length of the chord forming the segment. The formula for the Area Of A Segment Of A Circle depends on whether it is a minor or major segment and whether we use the central angle in degrees or radians. The Area Of A Segment Of A Circle increases as the central angle of the segment increases and can be used to find the area of a region bounded by two intersecting chords in a circle.

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Area Of A Segment Of A Circle

A segment of a circle is a region bounded by a chord and an arc of the circle. To find the area of a segment of a circle, we need to know the radius of the circle, the central angle (θ) of the segment, and the length of the chord (c) that forms the segment.

The area of a segment of a circle is a useful geometric concept that is used in various applications such as engineering, architecture, and physics. For example, in civil engineering, the area of a segment is used to calculate the volume of materials required for constructing arched structures, while in physics, the area of a segment is used to calculate the moment of inertia of a circular sector.

One interesting property of the area of a segment is that it increases as the central angle of the segment increases. This makes intuitive sense because as the central angle increases, the segment becomes closer to a sector of the circle, and the area of the segment approaches the area of the sector. Additionally, if the central angle of the segment is equal to 180 degrees, then the segment is a semicircle, and its area is half the area of the full circle.

The area of a segment can also be used to find the area of a region bounded by two intersecting chords in a circle. To do this, we can divide the region into two segments by drawing a line between the intersection point of the chords and the center of the circle. We can then find the area of each segment using the formulas mentioned above and add them together to obtain the total area of the region.

It is important to note that the formulas for the area of a segment of a circle assume that the chord forming the segment does not intersect the circle’s center. If the chord does intersect the center, then the segment becomes a triangle, and the area of the segment can be found using the formula for the area of a triangle instead.

Overall, the area of a segment of a circle is a fundamental concept in geometry with many practical applications. Its formulas allow us to calculate the area of various segments in a circle, which can be used in many different fields.

How Do I Find The Area Of A Segment?

To find the area of a segment of a circle, The area of a segment of a circle can also be calculated using the trigonometric functions sine, cosine, and tangent. To use this method, we need to know the radius of the circle, the central angle (θ) of the segment, and the length of the chord (c) that forms the segment. The formula for the area of a segment using trigonometry is:

Area of Segment = (θ/360)πr^2 – (1/2)c * sqrt(r^2 – (c^2/4))

where:

• θ is the central angle of the segment in degrees
• r is the radius of the circle
• c is the length of the chord that forms the segment

The formula consists of two parts. The first part calculates the area of the sector formed by the central angle of the segment, while the second part subtracts the area of the triangle formed by the chord and the two radii from the sector’s area to obtain the segment’s area.

What Is The Formula For Area Of Minor Segment?

A minor segment is a segment of a circle that lies between the chord and the minor arc (i.e., the smaller of the two arcs intercepted by the chord). The formula for the area of a minor segment of a circle is the same as that for the area of a segment, given as:

Area of Minor Segment = (θ/360)πr^2 – (1/2)c * sqrt(r^2 – (c^2/4))

where:

• θ is the central angle of the minor segment in degrees
• r is the radius of the circle
• c is the length of the chord that forms the minor segment

Note that for a minor segment, the central angle θ is less than 180 degrees, and the chord length c is less than the circle’s diameter. The formula for the area of a minor segment is the same as that for the area of a segment because the formula considers the area of the sector formed by the central angle θ and subtracts the area of the triangle formed by the chord and the radii.

The formula for the area of a major segment is derived by subtracting the area of the sector formed by the complement of the central angle θ (i.e., 2π – θ) from the area of the full circle, and then subtracting the area of the triangle formed by the chord and the radii.

In summary, the area of a segment of a circle can be calculated using different formulas depending on whether it is a minor or major segment and whether we prefer to use the central angle in degrees or radians. Regardless of the formula used, we need to know the radius of the circle and the length of the chord that forms the segment.

What Is The Formula Of Area Of Radius Segment?

The formula for the area of a segment of a circle formed by a radius and an arc is given by: Area of segment = (θ/2) × r² – ((1/2) × r² × sin(θ)) Where:

• θ is the central angle subtended by the segment in radians
• r is the radius of the circle

• The formula for the area of a segment of a circle with radius “r” and central angle “θ” is given by: Area of segment = (1/2) * r^2 * (θ – sinθ)
• Where “θ” is measured in radians, which can be converted from degrees by multiplying with π/180.
• For a radius segment, the central angle “θ” is 2π radians (or 360 degrees), and the formula reduces to: Area of radius segment = (1/2) * r^2 * (2π – sin(2π)) = (1/2) * r^2 * π
• Hence, the formula for the area of a radius segment is simply half of the area of the circle with radius “r”.

Area Of A Segment Of A Circle Without Angle

The formula for the area of a segment of a circle without the angle is not possible as the angle is a crucial parameter in determining the area of a segment of a circle. Without the angle, it is not possible to determine the area of the segment.

• If the central angle of a circle segment is not given, we can still find its area using the radius and height of the segment.
• The formula for the area of a circle segment with radius “r” and height “h” is given by: Area of segment = (r^2 / 2) * (sin⁻¹(h/r) – (h/r) * √(1 – (h/r)^2))
• Here, “sin⁻¹” represents the inverse sine function, and “√” represents the square root function.
• To find the height “h” of the segment, we can use the Pythagorean theorem, which states that the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse.
• In this case, the hypotenuse is the radius “r”, and the legs are half of the chord length “c” and the height “h”.
• Hence, we can find the height “h” using the formula: h = r – √(r^2 – (c/2)^2)
• Once we have the height “h”, we can use the formula above to find the area of the segment.

Area Of Minor Segment Formula

The formula for the area of a minor segment of a circle formed by a chord and an arc is given by: Area of minor segment = ((θ – sin(θ)) / 2) × r² Where:

• θ is the central angle subtended by the chord in radians
• r is the radius of the circle
• A minor segment of a circle is the region between a chord and the minor arc that it subtends.
• The formula for the area of a minor segment with radius “r”, chord length “c”, and central angle “θ” (measured in radians) is given by: Area of minor segment = (1/2) * r^2 * (θ – sinθ) – (1/2) * c * √(r^2 – (c/2)^2)
• Here, the first term represents the area of the sector (or wedge) formed by the central angle “θ”, and the second term represents the area of the triangular portion outside the sector.
• If the central angle “θ” is not given, we can find it using the chord length “c” and the radius “r” using the formula: θ = 2 * sin⁻¹(c / (2*r))
• Once we have the central angle “θ”, we can use the formula above to find the area of the minor segment.

Area Of A Segment Of A Circle – FAQs