Understanding Circles – Concepts, Formulas, and Applications

 

<h2>1. What is a Circle?</h2>

<p>A <strong>circle</strong> is the set of all points in a plane that are at a fixed distance (called the <strong>radius</strong>) from a fixed point (the <strong>centre</strong>).</p>


<ul>

  <li><strong>Centre</strong> = (h, k)</li>

  <li><strong>Radius</strong> = r</li>

  <li><strong>Diameter</strong> = d = 2r</li>

</ul>


<h2>2. Important Formulas</h2>

<ul>

  <li><strong>Circumference</strong>: C = 2 &pi; r</li>

  <li><strong>Area</strong>: A = &pi; r<sup>2</sup></li>

  <li><strong>Length of an arc</strong> (angle θ in degrees): arc = (θ / 360) × 2 &pi; r</li>

  <li><strong>Sector area</strong> (angle θ in degrees): sector = (θ / 360) × &pi; r<sup>2</sup></li>

  <li><strong>Equation of a circle</strong> with centre (h,k): (x − h)<sup>2</sup> + (y − k)<sup>2</sup> = r<sup>2</sup></li>

</ul>


<h2>3. Tangent, Chord & Secant</h2>

<ul>

  <li><strong>Tangent</strong>: a line that touches the circle at exactly one point. At the point of contact the tangent is perpendicular to the radius.</li>

  <li><strong>Chord</strong>: a line segment with both endpoints on the circle. The longest chord is the diameter.</li>

  <li><strong>Secant</strong>: a line that intersects the circle at two points.</li>

</ul>


<h2>4. Key Theorems & Properties</h2>

<ol>

  <li><strong>Angle in a semicircle</strong>: The angle subtended by a diameter at any point on the circle is 90&deg;.</li>

  <li><strong>Equal angles</strong>: Equal arcs subtend equal angles at the centre.</li>

  <li><strong>Tangent–Radius</strong>: Tangent ⟂ radius at point of contact.</li>

  <li><strong>Alternate segment theorem</strong>: Angle between tangent and chord equals angle in the opposite arc.</li>

</ol>


<h2>5. Sample Problems (with approach)</h2>


<p><strong>Example 1:</strong> Find the circumference and area of a circle of radius 7 cm.</p>

<p><strong>Solution (approach):</strong></p>

<ul>

  <li>Circumference = 2&pi;r = 2 &times; &pi; &times; 7 = 14&pi; cm</li>

  <li>Area = &pi;r<sup>2</sup> = &pi; &times; 7<sup>2</sup> = 49&pi; cm<sup>2</sup></li>

</ul>


<p><strong>Example 2:</strong> A sector of a circle with radius 10 cm subtends an angle of 60&deg;. Find the arc length and area of the sector.</p>

<p><strong>Approach:</strong></p>

<ul>

  <li>Arc length = (60/360) × 2&pi; × 10 = (1/6) × 20&pi; = (10/3)&pi; cm</li>

  <li>Sector area = (60/360) × &pi; × 10<sup>2</sup> = (1/6) × 100&pi; = (50/3)&pi; cm<sup>2</sup></li>

</ul>


<h2>6. Quick Tips for CET / Competitive Exams</h2>

<ul>

  <li>Memorize area & circumference formulas and common arc/sector fractions (30°, 60°, 90°).</li>

  <li>Always draw a clear diagram — many mistakes come from wrong sketches.</li>

  <li>Use &pi; ≈ 22/7 for simple exam calculations only when question allows.</li>

  <li>Remember: angle subtended by diameter = 90° (very common in questions).</li>

</ul>


<hr>


<p><em>Posted by: <strong>Sarthak Prakash Patil</strong></em></p>


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