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2026-01-01
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<p>Last updated on<strong>September 15, 2025</strong></p>
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<p>Last updated on<strong>September 15, 2025</strong></p>
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<p>The area of a segment is the region bounded by an arc and the chord joining the endpoints of the arc in a circle. Calculating the area of a segment involves understanding parts of the circle and applying the correct formulas. In this section, we will explore how to find the area of a segment.</p>
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<p>The area of a segment is the region bounded by an arc and the chord joining the endpoints of the arc in a circle. Calculating the area of a segment involves understanding parts of the circle and applying the correct formulas. In this section, we will explore how to find the area of a segment.</p>
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<h2>What is the Area of a Segment?</h2>
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<h2>What is the Area of a Segment?</h2>
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<p>A segment is a part<a>of</a>a circle formed by an arc and the chord connecting the arc's endpoints. The area of a segment is the space enclosed within these boundaries.</p>
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<p>A segment is a part<a>of</a>a circle formed by an arc and the chord connecting the arc's endpoints. The area of a segment is the space enclosed within these boundaries.</p>
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<p>To find the area of a segment, you'll need to understand both the sector of the circle formed by the arc and the triangular area formed by the chord.</p>
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<p>To find the area of a segment, you'll need to understand both the sector of the circle formed by the arc and the triangular area formed by the chord.</p>
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<h2>Area of the Segment Formula</h2>
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<h2>Area of the Segment Formula</h2>
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<p>To find the area of a segment, we use the<a>formula</a>: Segment Area = Sector Area - Triangle Area. The sector area is given by (θ/360) × π × r², where θ is the central angle in degrees and r is the radius. The triangle area can be calculated using various methods, such as<a>trigonometry</a>or coordinate<a>geometry</a>.</p>
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<p>To find the area of a segment, we use the<a>formula</a>: Segment Area = Sector Area - Triangle Area. The sector area is given by (θ/360) × π × r², where θ is the central angle in degrees and r is the radius. The triangle area can be calculated using various methods, such as<a>trigonometry</a>or coordinate<a>geometry</a>.</p>
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<p>Derivation of the formula: 1. Calculate the sector area using the formula: (θ/360) × π × r². 2. Find the area of the triangle formed by the chord and the radii. This can be done using trigonometric identities if the angle is known. 3. Subtract the triangle area from the sector area to get the segment area.</p>
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<p>Derivation of the formula: 1. Calculate the sector area using the formula: (θ/360) × π × r². 2. Find the area of the triangle formed by the chord and the radii. This can be done using trigonometric identities if the angle is known. 3. Subtract the triangle area from the sector area to get the segment area.</p>
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<h2>How to Find the Area of a Segment?</h2>
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<h2>How to Find the Area of a Segment?</h2>
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<p>We can find the area of a segment using different methods, depending on the available information. Here are some methods: Method using the central angle and radius Method using the chord length and radius Method using the arc length Now let’s discuss these methods.</p>
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<p>We can find the area of a segment using different methods, depending on the available information. Here are some methods: Method using the central angle and radius Method using the chord length and radius Method using the arc length Now let’s discuss these methods.</p>
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<p><strong>Method Using the Central Angle and Radius</strong></p>
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<p><strong>Method Using the Central Angle and Radius</strong></p>
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<p>If the central angle (θ) and radius (r) are given, find the sector area and then subtract the triangle area. For example, if θ is 60 degrees and r is 10 cm: </p>
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<p>If the central angle (θ) and radius (r) are given, find the sector area and then subtract the triangle area. For example, if θ is 60 degrees and r is 10 cm: </p>
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<p>Sector Area = (θ/360) × π × r² = (60/360) × π × 10² = (1/6) × π × 100 = 52.36 cm² </p>
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<p>Sector Area = (θ/360) × π × r² = (60/360) × π × 10² = (1/6) × π × 100 = 52.36 cm² </p>
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<p>Triangle Area = 0.5 × r² × sin(θ) = 0.5 × 10² × sin(60) = 43.30 cm²</p>
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<p>Triangle Area = 0.5 × r² × sin(θ) = 0.5 × 10² × sin(60) = 43.30 cm²</p>
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<p>Segment Area = Sector Area - Triangle Area = 52.36 - 43.30 = 9.06 cm²</p>
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<p>Segment Area = Sector Area - Triangle Area = 52.36 - 43.30 = 9.06 cm²</p>
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<p><strong>Method Using the Chord Length and Radius</strong></p>
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<p><strong>Method Using the Chord Length and Radius</strong></p>
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<p>If the chord length (c) and radius (r) are given, use trigonometry to find θ and proceed as before. For instance, if c is 12 cm and r is 10 cm: 1. Find θ using cos(θ/2) = c/(2r) 2. Calculate the sector and triangle areas as above.</p>
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<p>If the chord length (c) and radius (r) are given, use trigonometry to find θ and proceed as before. For instance, if c is 12 cm and r is 10 cm: 1. Find θ using cos(θ/2) = c/(2r) 2. Calculate the sector and triangle areas as above.</p>
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<p><strong>Method Using the Arc Length</strong></p>
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<p><strong>Method Using the Arc Length</strong></p>
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<p>If the arc length (l) and radius (r) are given, find θ using θ = (l/r), then proceed with the previous methods.</p>
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<p>If the arc length (l) and radius (r) are given, find θ using θ = (l/r), then proceed with the previous methods.</p>
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<h2>Unit of Area of Segment</h2>
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<h2>Unit of Area of Segment</h2>
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<p>The area of a segment is measured in<a>square</a>units. The<a>measurement</a>depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).</p>
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<p>The area of a segment is measured in<a>square</a>units. The<a>measurement</a>depends on the system used: In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).</p>
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<p>In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).</p>
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<p>In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).</p>
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<h2>Special Cases or Variations for the Area of Segment</h2>
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<h2>Special Cases or Variations for the Area of Segment</h2>
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<p>Since a segment is part of a circle, its area can be computed using different methods based on available<a>data</a>. Consider these special cases:</p>
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<p>Since a segment is part of a circle, its area can be computed using different methods based on available<a>data</a>. Consider these special cases:</p>
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<p><strong>Case 1:</strong>Use of Central Angle and Radius If the central angle and radius are known, compute the segment area by finding the sector and triangle areas.</p>
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<p><strong>Case 1:</strong>Use of Central Angle and Radius If the central angle and radius are known, compute the segment area by finding the sector and triangle areas.</p>
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<p><strong>Case 2:</strong>Use of Chord Length and Radius If the chord length and radius are provided, use trigonometry to find the central angle, then compute the segment area.</p>
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<p><strong>Case 2:</strong>Use of Chord Length and Radius If the chord length and radius are provided, use trigonometry to find the central angle, then compute the segment area.</p>
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<p><strong>Case 3:</strong>Using Arc Length If the arc length is known, derive the central angle and apply the standard segment area formula.</p>
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<p><strong>Case 3:</strong>Using Arc Length If the arc length is known, derive the central angle and apply the standard segment area formula.</p>
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<h2>Tips and Tricks for Area of Segment</h2>
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<h2>Tips and Tricks for Area of Segment</h2>
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<p>To ensure accurate results when calculating the area of a segment, keep these tips in mind:</p>
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<p>To ensure accurate results when calculating the area of a segment, keep these tips in mind:</p>
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<ul><li>Always verify the unit of measurement and convert if necessary. </li>
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<ul><li>Always verify the unit of measurement and convert if necessary. </li>
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<li>Understand the difference between sector area and segment area. </li>
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<li>Understand the difference between sector area and segment area. </li>
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<li>Use trigonometric identities effectively when angles or side lengths are given. </li>
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<li>Use trigonometric identities effectively when angles or side lengths are given. </li>
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<li>Don't forget to subtract the triangle area from the sector area to get the segment area.</li>
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<li>Don't forget to subtract the triangle area from the sector area to get the segment area.</li>
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</ul><h2>Common Mistakes and How to Avoid Them in Area of Segment</h2>
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</ul><h2>Common Mistakes and How to Avoid Them in Area of Segment</h2>
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<p>It's common for students to make mistakes while finding the area of a segment. Let’s explore some common errors and how to avoid them.</p>
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<p>It's common for students to make mistakes while finding the area of a segment. Let’s explore some common errors and how to avoid them.</p>
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<h3>Problem 1</h3>
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<h3>Problem 1</h3>
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<p>The central angle of a segment in a circle with a radius of 10 cm is given as 90 degrees. What will be the area of the segment?</p>
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<p>The central angle of a segment in a circle with a radius of 10 cm is given as 90 degrees. What will be the area of the segment?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We will find the area as 21.46 cm².</p>
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<p>We will find the area as 21.46 cm².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>With a central angle of 90 degrees and radius of 10 cm, calculate the sector area and subtract the triangle area: </p>
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<p>With a central angle of 90 degrees and radius of 10 cm, calculate the sector area and subtract the triangle area: </p>
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<p>Sector Area = (90/360) × π × 10² = 78.54 cm² </p>
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<p>Sector Area = (90/360) × π × 10² = 78.54 cm² </p>
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<p>Triangle Area = 0.5 × 10² × sin(90) = 50 cm² </p>
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<p>Triangle Area = 0.5 × 10² × sin(90) = 50 cm² </p>
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<p>Segment Area = Sector Area - Triangle Area = 78.54 - 50 = 28.54 cm²</p>
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<p>Segment Area = Sector Area - Triangle Area = 78.54 - 50 = 28.54 cm²</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 2</h3>
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<h3>Problem 2</h3>
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<p>A segment has a chord length of 8 cm and a radius of 5 cm. What is the area of the segment?</p>
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<p>A segment has a chord length of 8 cm and a radius of 5 cm. What is the area of the segment?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We will find the area as 3.94 cm².</p>
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<p>We will find the area as 3.94 cm².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Given a chord length of 8 cm and radius of 5 cm, use trigonometry to find the central angle, then compute the segment area.</p>
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<p>Given a chord length of 8 cm and radius of 5 cm, use trigonometry to find the central angle, then compute the segment area.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 3</h3>
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<h3>Problem 3</h3>
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<p>The arc length of a segment is 6 cm, and the radius is 4 cm. What is the area of the segment?</p>
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<p>The arc length of a segment is 6 cm, and the radius is 4 cm. What is the area of the segment?</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We find the area of the segment as 4.19 cm².</p>
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<p>We find the area of the segment as 4.19 cm².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Find the central angle using θ = (l/r), then calculate the sector and triangle areas.</p>
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<p>Find the central angle using θ = (l/r), then calculate the sector and triangle areas.</p>
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<p>Subtract the triangle area from the sector area to find the segment area.</p>
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<p>Subtract the triangle area from the sector area to find the segment area.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 4</h3>
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<h3>Problem 4</h3>
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<p>Find the area of the segment in a circle with a radius of 7 cm and a central angle of 45 degrees.</p>
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<p>Find the area of the segment in a circle with a radius of 7 cm and a central angle of 45 degrees.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We will find the area as 4.51 cm².</p>
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<p>We will find the area as 4.51 cm².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>With a radius of 7 cm and central angle of 45 degrees, calculate:</p>
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<p>With a radius of 7 cm and central angle of 45 degrees, calculate:</p>
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<p>Sector Area = (45/360) × π × 7² = 19.24 cm²</p>
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<p>Sector Area = (45/360) × π × 7² = 19.24 cm²</p>
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<p>Triangle Area = 0.5 × 7² × sin(45) = 14.73 cm²</p>
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<p>Triangle Area = 0.5 × 7² × sin(45) = 14.73 cm²</p>
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<p>Segment Area = Sector Area - Triangle Area = 19.24 - 14.73 = 4.51 cm²</p>
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<p>Segment Area = Sector Area - Triangle Area = 19.24 - 14.73 = 4.51 cm²</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h3>Problem 5</h3>
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<h3>Problem 5</h3>
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<p>Help Sarah find the area of a segment if the radius is 6 m and the chord length is 10 m.</p>
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<p>Help Sarah find the area of a segment if the radius is 6 m and the chord length is 10 m.</p>
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<p>Okay, lets begin</p>
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<p>Okay, lets begin</p>
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<p>We will find the area as 6.28 m².</p>
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<p>We will find the area as 6.28 m².</p>
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<h3>Explanation</h3>
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<h3>Explanation</h3>
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<p>Use the chord length and radius to find the central angle, then calculate the segment area by subtracting the triangle area from the sector area.</p>
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<p>Use the chord length and radius to find the central angle, then calculate the segment area by subtracting the triangle area from the sector area.</p>
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<p>Well explained 👍</p>
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<p>Well explained 👍</p>
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<h2>FAQs on Area of Segment</h2>
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<h2>FAQs on Area of Segment</h2>
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<h3>1.Can the area of a segment be negative?</h3>
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<h3>1.Can the area of a segment be negative?</h3>
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<p>No, the area of a segment cannot be negative. The area of any shape is always positive.</p>
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<p>No, the area of a segment cannot be negative. The area of any shape is always positive.</p>
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<h3>2.How to find the area of a segment if the central angle and radius are given?</h3>
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<h3>2.How to find the area of a segment if the central angle and radius are given?</h3>
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<p>Calculate the sector area using (θ/360) × π × r² and subtract the triangle area using 0.5 × r² × sin(θ).</p>
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<p>Calculate the sector area using (θ/360) × π × r² and subtract the triangle area using 0.5 × r² × sin(θ).</p>
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<h3>3.How to find the area of a segment if only the chord length and radius are given?</h3>
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<h3>3.How to find the area of a segment if only the chord length and radius are given?</h3>
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<p>Use trigonometry to find the central angle, then apply the segment area formula.</p>
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<p>Use trigonometry to find the central angle, then apply the segment area formula.</p>
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<h3>4.What is the difference between the area of a sector and the area of a segment?</h3>
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<h3>4.What is the difference between the area of a sector and the area of a segment?</h3>
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<p>The area of a sector includes the triangular part formed by the chord, while the area of a segment excludes it.</p>
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<p>The area of a sector includes the triangular part formed by the chord, while the area of a segment excludes it.</p>
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<h3>5.How is the chord length used in segment calculations?</h3>
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<h3>5.How is the chord length used in segment calculations?</h3>
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<p>The chord length helps determine the triangle area within the segment, which is subtracted from the sector area to find the segment area.</p>
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<p>The chord length helps determine the triangle area within the segment, which is subtracted from the sector area to find the segment area.</p>
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<h2>Seyed Ali Fathima S</h2>
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<h2>Seyed Ali Fathima S</h2>
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<h3>About the Author</h3>
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<h3>About the Author</h3>
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<p>Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.</p>
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<p>Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.</p>
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<h3>Fun Fact</h3>
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<h3>Fun Fact</h3>
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<p>: She has songs for each table which helps her to remember the tables</p>
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<p>: She has songs for each table which helps her to remember the tables</p>