Circles, Ellipses & Conic Sections | Free ACT Math Course

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<title>ACT Math: Circles, Ellipses & Conic Sections | The School of Mathematics</title>

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<h1>ACT Math: Circles, Ellipses & Conic Sections</h1>

Conic sections make up a small but very learnable slice of the ACT Math test, and circles do the vast majority of the work. This free, complete lesson covers the equation of a circle (including finding the center and radius by completing the square), arc length and sector area, central and inscribed angles with tangent lines, and ellipses along with recognizing the different conic sections. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.

</p>

On the ACT, conic section questions overwhelmingly focus on circles. Ellipses appear only occasionally, usually just asking you to recognize or interpret the standard equation, and hyperbolas essentially never appear, so this lesson doesn't cover them.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-circles-ellipses-conic-sections-quiz-1">Practice Circles, Ellipses & Conic Sections Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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<h2>What's covered in this lesson</h2>

<ol>

<li><a href="#equation">The Equation of a Circle</a></li>

<li><a href="#arc-sector">Arc Length and Sector Area</a></li>

<li><a href="#angles">Central and Inscribed Angles, Tangent Lines</a></li>

<li><a href="#ellipses">Ellipses and Recognizing Conic Sections</a></li>

<li><a href="#mistakes">Common Mistakes to Avoid</a></li>

<li><a href="#faq">Frequently Asked Questions</a></li>

</ol>

</nav>

<h2 id="equation">1. The Equation of a Circle</h2>

<p>The equation of a circle with center (h, k) and radius r is:</p>

<p>If given in general (expanded) form, complete the square on the x-terms and y-terms separately to rewrite it in this center-radius form.</p>

<p><strong>Worked Example:</strong> Find the center and radius of (x − 4)² + (y + 2)² = 36.</p>

<p>The center is (4, −2) and the radius is √36 = 6.</p>

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<p class="prompt">1. What is the center and radius of (x − 3)² + (y − 5)² = 49?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pa-1',true)">A) Center (3, 5), radius 7</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">B) Center (−3, −5), radius 7</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">C) Center (3, 5), radius 49</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pa-1',false)">D) Center (5, 3), radius 7</button>

</div>

<p><span class="label">Explanation:</span> Comparing to (x − h)² + (y − k)² = r²: h = 3, k = 5, and r² = 49, so r = 7.</p>

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<p class="prompt">2. What is the equation of a circle with center (−2, 6) and radius 4?</p>

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<p><span class="label">Explanation:</span> With h = −2 and k = 6: (x − (−2))² + (y − 6)² = 4², giving (x + 2)² + (y − 6)² = 16.</p>

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<p class="prompt">3. A circle has equation x² + y² = 64. What is its radius?</p>

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<p><span class="label">Explanation:</span> This circle is centered at the origin with r² = 64, so r = √64 = 8.</p>

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<p class="prompt">4. A circle is given by x² + y² − 4x + 6y + 4 = 0. What is the center?</p>

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<p><span class="label">Explanation:</span> Completing the square: (x² − 4x + 4) + (y² + 6y + 9) = −4 + 4 + 9, giving (x − 2)² + (y + 3)² = 9. The center is (2, −3).</p>

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<p class="prompt">5. Using the equation from the previous problem, what is the radius?</p>

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<p><span class="label">Explanation:</span> From (x − 2)² + (y + 3)² = 9, r² = 9, so r = 3.</p>

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<h2 id="arc-sector">2. Arc Length and Sector Area</h2>

<p class="step-math" style="text-align:center; font-size:1.05rem;">arc length = <span class="frac"><span class="num">θ</span><span class="den">360</span></span> × 2πr       sector area = <span class="frac"><span class="num">θ</span><span class="den">360</span></span> × πr²</p>

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<p class="figure-caption">A sector with central angle θ</p>

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<p><strong>Worked Example:</strong> A circle has radius 12. Find the arc length of a 60° arc.</p>

<p class="step-math">arc length = <span class="frac"><span class="num">60</span><span class="den">360</span></span> × 2π(12) = <span class="frac"><span class="num">1</span><span class="den">6</span></span> × 24π = 4π</p>

</div>

<p class="prompt">1. A circle has radius 9. What is the arc length of a 40° arc, in terms of π?</p>

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<p><span class="label">Explanation:</span> arc length = <span class="frac"><span class="num">40</span><span class="den">360</span></span> × 2π(9) = <span class="frac"><span class="num">1</span><span class="den">9</span></span> × 18π = 2π.</p>

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<p class="prompt">2. A circle has radius 10. What is the area of a 90° sector, in terms of π?</p>

</div>

<p><span class="label">Explanation:</span> sector area = <span class="frac"><span class="num">90</span><span class="den">360</span></span> × π(10)² = <span class="frac"><span class="num">1</span><span class="den">4</span></span> × 100π = 25π.</p>

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<p class="prompt">3. A circle has radius 6. An arc has length 2π. What is the central angle, in degrees?</p>

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<p><span class="label">Explanation:</span> 2π = <span class="frac"><span class="num">θ</span><span class="den">360</span></span> × 12π, so <span class="frac"><span class="num">θ</span><span class="den">360</span></span> = <span class="frac"><span class="num">1</span><span class="den">6</span></span>, giving θ = 60°.</p>

</div>

<p class="prompt">4. A sector of a circle with radius 8 has an area of 16π. What is the central angle, in degrees?</p>

</div>

<p><span class="label">Explanation:</span> 16π = <span class="frac"><span class="num">θ</span><span class="den">360</span></span> × 64π, so <span class="frac"><span class="num">θ</span><span class="den">360</span></span> = <span class="frac"><span class="num">1</span><span class="den">4</span></span>, giving θ = 90°.</p>

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<p class="prompt">5. A pizza with radius 7 inches is cut into a slice with a 45° central angle. What is the area of the slice, in terms of π?</p>

</div>

<p><span class="label">Explanation:</span> sector area = <span class="frac"><span class="num">45</span><span class="den">360</span></span> × 49π = <span class="frac"><span class="num">1</span><span class="den">8</span></span> × 49π = <span class="frac"><span class="num">49π</span><span class="den">8</span></span>.</p>

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<h2 id="angles">3. Central and Inscribed Angles, Tangent Lines</h2>

<p>A <strong>central angle</strong> has its vertex at the circle's center and equals its intercepted arc. An <strong>inscribed angle</strong> has its vertex on the circle and equals half its intercepted arc. A <strong>tangent</strong> line touching a circle at one point is always perpendicular to the radius drawn to that point.</p>

<tr><th>Rule</th><th>Statement</th></tr>

<tr><td>Central angle</td><td>Equals the measure of its intercepted arc</td></tr>

<tr><td>Inscribed angle</td><td>Equals half its intercepted arc</td></tr>

<tr><td>Inscribed angle on a diameter</td><td>Always 90°</td></tr>

<tr><td>Tangent-radius</td><td>Always perpendicular (90°)</td></tr>

<tr><td>Cyclic quadrilateral</td><td>Opposite angles are supplementary</td></tr>

</table>

<p><strong>Worked Example:</strong> A central angle intercepts an 80° arc. Find the inscribed angle intercepting the same arc.</p>

<p class="step-math">inscribed angle = 80° ÷ 2 = 40°</p>

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<p class="prompt">1. A central angle measures 100°. What is the inscribed angle intercepting the same arc?</p>

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<p><span class="label">Explanation:</span> An inscribed angle is half its central angle: 100° ÷ 2 = 50°.</p>

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<p class="prompt">2. An inscribed angle measures 35°. What is the central angle intercepting the same arc?</p>

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<p><span class="label">Explanation:</span> The central angle is double the inscribed angle: 35° × 2 = 70°.</p>

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<p class="prompt">3. An inscribed angle intercepts a semicircle (a 180° arc). What is the measure of the inscribed angle?</p>

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<p><span class="label">Explanation:</span> 180° ÷ 2 = 90°. Any inscribed angle that intercepts a diameter is always a right angle.</p>

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<p class="prompt">4. A line is tangent to a circle at point P. What is the angle between the tangent line and the radius drawn to P?</p>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pc-4',false)">D) Depends on the radius</button>

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<p><span class="label">Explanation:</span> A tangent line is always perpendicular to the radius at the point of tangency, regardless of the circle's size.</p>

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<p class="prompt">5. A cyclic quadrilateral is inscribed in a circle. One angle measures 75°. What is the measure of its opposite angle?</p>

</div>

<p><span class="label">Explanation:</span> Opposite angles of a quadrilateral inscribed in a circle are always supplementary: 180° − 75° = 105°.</p>

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<h2 id="ellipses">4. Ellipses and Recognizing Conic Sections</h2>

<p>An <strong>ellipse</strong> centered at (h, k) has the standard-form equation:</p>

<p>Here a is the horizontal semi-axis length and b is the vertical semi-axis length. If a = b, the equation actually describes a circle.</p>

<tr><th>Conic section</th><th>How to recognize it</th></tr>

<tr><td>Circle</td><td>Both variables squared, same coefficient</td></tr>

<tr><td>Ellipse</td><td>Both variables squared, different (unequal) coefficients, same sign</td></tr>

<tr><td>Parabola</td><td>Only one variable squared</td></tr>

</table>

<p><strong>Worked Example:</strong> An ellipse has equation <span class="frac"><span class="num">x²</span><span class="den">25</span></span> + <span class="frac"><span class="num">y²</span><span class="den">9</span></span> = 1. Find the length of the horizontal and vertical axes.</p>

<p>Horizontal axis length: 2a = 10. Vertical axis length: 2b = 6.</p>

</div>

<p class="prompt">1. An ellipse has equation <span class="frac"><span class="num">x²</span><span class="den">16</span></span> + <span class="frac"><span class="num">y²</span><span class="den">4</span></span> = 1. What is the length of the horizontal axis?</p>

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<p><span class="label">Explanation:</span> a² = 16, so a = 4. Horizontal axis length: 2a = 8.</p>

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<p class="prompt">2. Using the same ellipse, what is the length of the vertical axis?</p>

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<p><span class="label">Explanation:</span> b² = 4, so b = 2. Vertical axis length: 2b = 4.</p>

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<p class="prompt">3. What is the center of the ellipse <span class="frac"><span class="num">(x − 3)²</span><span class="den">25</span></span> + <span class="frac"><span class="num">(y + 2)²</span><span class="den">9</span></span> = 1?</p>

</div>

<p><span class="label">Explanation:</span> Since (y + 2)² = (y − (−2))², the center is (3, −2).</p>

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<p class="prompt">4. Which equation represents a circle rather than an ellipse?</p>

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<p><span class="label">Explanation:</span> When a² = b² (here, both equal 9), the equation describes a circle rather than a true (non-circular) ellipse.</p>

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<p class="prompt">5. Which conic section has an equation with only one squared variable?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-5',true)">A) Parabola</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-5',false)">B) Circle</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-5',false)">C) Ellipse</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-5',false)">D) All conics have two squared variables</button>

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<p><span class="label">Explanation:</span> A parabola's equation, like y = ax² + bx + c, has only one squared variable, while circles and ellipses both have two.</p>

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<h2 id="mistakes">Common Mistakes to Avoid</h2>

<li><strong>Reading the center coordinates with the wrong sign.</strong> In (x − h)² + (y − k)² = r², an expression like (x + 4)² means h = −4, not 4.</li>

<li><strong>Forgetting to take the square root when finding the radius.</strong> The right side of the circle equation is r², not r. If the equation ends in "= 25," the radius is 5.</li>

<li><strong>Mixing up the degree-based arc length and sector area formulas' denominator.</strong> Both use <span class="frac"><span class="num">θ</span><span class="den">360</span></span> as the fraction of the full circle, don't forget this factor before multiplying by circumference or area.</li>

<li><strong>Assuming an inscribed angle equals its central angle instead of half of it.</strong> An inscribed angle is always half the central angle intercepting the same arc, not equal to it.</li>

<li><strong>Confusing an ellipse's semi-axis (a or b) with its full axis length.</strong> The value under the denominator gives a² or b²; take the square root, then double it, to get the full axis length.</li>

</ul>

<h2 id="faq">Frequently Asked Questions</h2>

<h3>How many conic section questions typically appear on the ACT?</h3>

<p>Usually just 1 to 2 questions, and they overwhelmingly focus on circle equations. Spending the bulk of your prep time mastering the circle equation, rather than splitting focus evenly across every conic type, is the more efficient strategy.</p>

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<h3>Do I need to memorize the equation for an ellipse's foci?</h3>

<p>No, ACT ellipse questions typically stop at recognizing the standard form and reading off the center and axis lengths. Foci calculations go beyond what's realistically tested.</p>

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<h3>How can I quickly tell a circle equation from an ellipse equation?</h3>

<p>Check whether the two denominators under the squared terms are equal. If they match, it's a circle. If they're different (but both positive, with a plus sign between the terms), it's an ellipse.</p>

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<h3>Where can I practice more problems like these?</h3>

<p>The <a href="https://theschoolofmathematics.com/quiz/act-circles-ellipses-conic-sections-quiz-1">Circles, Ellipses & Conic Sections quizzes</a> in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.</p>

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-circles-ellipses-conic-sections-quiz-1">Practice Circles, Ellipses & Conic Sections Free</a>

<a class="cta-btn cta-secondary" href="https://theschoolofmathematics.com/quiz/course/ACT-Math-Qbank">Explore the Full ACT Math Qbank</a>

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