<h3>The Main Idea</h3> <p><strong>Know what you want, know what you have, pick the right formula.</strong></p> <p>Circle problems come down to two measurements: circumference (distance around) and area (space inside). Each has its own formula, and which one you use depends on whether you're given the radius or diameter.</p> <h3>Radius vs. Diameter</h3> <p>Before touching any formula, make sure you know the difference:</p> <ul> <li><strong>Radius (r)</strong> — from the center to the edge (the smaller one)</li> <li><strong>Diameter (d)</strong> — all the way across through the center (the bigger one)</li> <li><strong>Relationship:</strong> Diameter = 2 × Radius, or Radius = Diameter ÷ 2</li> </ul> <h3>Circumference (Distance Around)</h3> <p>Two formulas — pick based on what you're given:</p> <ul> <li>Given radius: <strong>C = 2πr</strong></li> <li>Given diameter: <strong>C = πd</strong></li> </ul> <p>Both give the same answer. Use whichever matches what the problem gives you.</p> <h3>Area (Space Inside)</h3> <p>One formula: <strong>A = πr²</strong></p> <p><strong>Watch out:</strong> Squared means multiply by itself (8² = 64), NOT double (8 × 2 = 16). This is the most common mistake.</p> <h3>π: Leave It or Calculate?</h3> <p>Check the answer choices:</p> <ul> <li>If answers have π in them (like 20π) → leave π as π</li> <li>If answers are plain numbers (like 157) → use 3.14 for π</li> </ul> <h3>Working Backwards</h3> <p>Sometimes they give you circumference or area and ask for radius. Just work the formula backwards:</p> <ul> <li>From circumference: r = C ÷ 2π</li> <li>From area: r² = A ÷ π, then take the square root</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Sectors and arcs are just fractions of circles.</strong></p> <p>If you know how to find circumference and area of a full circle, you can find the arc length and sector area — just take the right fraction based on the angle.</p> <h3>The Fraction Formula</h3> <p>Every sector/arc problem comes down to one question: <strong>What fraction of the circle is this?</strong></p> $$\text{Fraction} = \frac{\theta}{360°}$$ <p>where θ is the central angle in degrees.</p> <h3>Arc Length (Distance Along the Curve)</h3> <p>An arc is part of the circumference. Take the fraction of the full circumference:</p> $$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$ <h3>Sector Area (Pizza Slice)</h3> <p>A sector is a "pizza slice" of the circle. Take the fraction of the full area:</p> $$\text{Sector Area} = \frac{\theta}{360} \times \pi r^2$$ <h3>Sector Perimeter (Going Around the Wedge)</h3> <p>The perimeter of a sector includes THREE parts:</p> <ul> <li>The curved arc</li> <li>Two straight edges (both are radii)</li> </ul> $$\text{Sector Perimeter} = \text{Arc Length} + 2r$$ <p><strong>Special case:</strong> For a semicircle (180°), the two radii form a diameter, so: Arc + Diameter</p> <h3>Common Angles to Know</h3> <ul> <li>180° = 1/2 of circle (semicircle)</li> <li>90° = 1/4 of circle (quarter)</li> <li>60° = 1/6 of circle</li> <li>45° = 1/8 of circle</li> </ul>
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