Perimeter, Area & Volume | Free ACT Math Course

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<h1>ACT Math: Perimeter, Area & Volume</h1>

Perimeter, Area & Volume questions on the ACT come down to memorizing about ten formulas and applying them carefully. This free, complete lesson covers perimeter and area of 2D shapes, circles (circumference and area), volume of 3D solids like prisms, cylinders, cones, spheres, and pyramids, and surface area with composite figures. 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.

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The ACT provides no general formula sheet for the Math section, though occasionally a less common formula (like a cone's volume) is given directly within a specific question. Knowing the core formulas by heart is worth the time investment.

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<a class="cta-btn cta-primary" href="https://theschoolofmathematics.com/quiz/act-perimeter-area-volume-quiz-1">Practice Perimeter, Area & Volume 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="#area-2d">Perimeter and Area of 2D Shapes</a></li>

<li><a href="#circles">Circles: Circumference and Area</a></li>

<li><a href="#volume">Volume of 3D Solids</a></li>

<li><a href="#surface-area">Surface Area and Composite Figures</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="area-2d">1. Perimeter and Area of 2D Shapes</h2>

<tr><th>Shape</th><th>Area formula</th></tr>

<tr><td>Rectangle</td><td>A = lw</td></tr>

<tr><td>Triangle</td><td>A = <span class="frac"><span class="num">1</span><span class="den">2</span></span>bh</td></tr>

<tr><td>Parallelogram</td><td>A = bh</td></tr>

<tr><td>Trapezoid</td><td>A = <span class="frac"><span class="num">1</span><span class="den">2</span></span>(b<sub>1</sub> + b<sub>2</sub>)h</td></tr>

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<p><strong>Worked Example:</strong> A rectangle has length 12 and width 7. Find its area and perimeter.</p>

<p class="step-math">Perimeter = 2(12 + 7) = 38</p>

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<p class="prompt">1. A triangle has base 14 and height 9. What is its area?</p>

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<p><span class="label">Explanation:</span> A = <span class="frac"><span class="num">1</span><span class="den">2</span></span>(14)(9) = 63.</p>

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<p class="prompt">2. A parallelogram has base 11 and height 6. What is its area?</p>

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<p><span class="label">Explanation:</span> A = bh = 11 × 6 = 66.</p>

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<p class="prompt">3. A trapezoid has parallel sides of 8 and 14, and a height of 5. What is its area?</p>

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<p><span class="label">Explanation:</span> A = <span class="frac"><span class="num">1</span><span class="den">2</span></span>(8 + 14)(5) = <span class="frac"><span class="num">1</span><span class="den">2</span></span>(22)(5) = 55.</p>

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<p class="prompt">4. A rectangle has a perimeter of 34 and a length of 10. What is its area?</p>

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<p><span class="label">Explanation:</span> Perimeter = 2(l + w) = 34, so l + w = 17. With l = 10, w = 7. Area = 10 × 7 = 70.</p>

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<p class="prompt">5. A square has a perimeter of 36. What is its area?</p>

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<p><span class="label">Explanation:</span> Each side is 36 ÷ 4 = 9. Area = 9² = 81.</p>

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<h2 id="circles">2. Circles: Circumference and Area</h2>

<p><strong>Worked Example:</strong> A circle has radius 6. Find its circumference and area.</p>

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<p class="prompt">1. A circle has radius 9. What is its circumference, in terms of π?</p>

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<p><span class="label">Explanation:</span> C = 2π(9) = 18π.</p>

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<p class="prompt">2. A circle has radius 9. What is its area, in terms of π?</p>

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<p><span class="label">Explanation:</span> A = π(9)² = 81π.</p>

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<p class="prompt">3. A circle has a diameter of 20. What is its area, in terms of π?</p>

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<p><span class="label">Explanation:</span> The radius is half the diameter: 10. A = π(10)² = 100π.</p>

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<p class="prompt">4. A circle has a circumference of 20π. What is its radius?</p>

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<p><span class="label">Explanation:</span> 2πr = 20π, so r = 10.</p>

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<p class="prompt">5. A circle has an area of 49π. What is its diameter?</p>

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<p><span class="label">Explanation:</span> πr² = 49π, so r² = 49 and r = 7. The diameter is twice the radius: 14.</p>

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<h2 id="volume">3. Volume of 3D Solids</h2>

<tr><th>Solid</th><th>Volume formula</th></tr>

<tr><td>Rectangular prism</td><td>V = lwh</td></tr>

<tr><td>Cylinder</td><td>V = πr²h</td></tr>

<tr><td>Sphere</td><td>V = <span class="frac"><span class="num">4</span><span class="den">3</span></span>πr³</td></tr>

<tr><td>Pyramid</td><td>V = <span class="frac"><span class="num">1</span><span class="den">3</span></span>Bh   (B = base area)</td></tr>

</table>

<p><strong>Worked Example:</strong> A rectangular prism has length 5, width 4, and height 3. Find its volume.</p>

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<p class="prompt">1. A rectangular prism has length 8, width 5, and height 3. What is its volume?</p>

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<p><span class="label">Explanation:</span> V = 8 × 5 × 3 = 120.</p>

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<p class="prompt">2. A cylinder has radius 4 and height 10. What is its volume, in terms of π?</p>

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<p><span class="label">Explanation:</span> V = π(4)²(10) = 160π.</p>

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<p class="prompt">3. A cube has an edge length of 6. What is its volume?</p>

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<p><span class="label">Explanation:</span> V = 6³ = 216.</p>

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<p class="prompt">4. A cone has radius 3 and height 8. What is its volume, in terms of π?</p>

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<p><span class="label">Explanation:</span> V = <span class="frac"><span class="num">1</span><span class="den">3</span></span>π(3)²(8) = <span class="frac"><span class="num">1</span><span class="den">3</span></span>π(9)(8) = <span class="frac"><span class="num">1</span><span class="den">3</span></span>(72π) = 24π.</p>

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<p class="prompt">5. A sphere has radius 3. What is its volume, in terms of π?</p>

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<p><span class="label">Explanation:</span> V = <span class="frac"><span class="num">4</span><span class="den">3</span></span>π(3)³ = <span class="frac"><span class="num">4</span><span class="den">3</span></span>π(27) = 36π.</p>

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<p class="prompt">6. A pyramid has a square base with side length 6 and a height of 10. What is its volume?</p>

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<p><span class="label">Explanation:</span> The base area is 6² = 36. V = <span class="frac"><span class="num">1</span><span class="den">3</span></span>(36)(10) = 120.</p>

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<h2 id="surface-area">4. Surface Area and Composite Figures</h2>

<p><strong>Surface area</strong> measures the total area covering a solid's exterior. A <strong>composite figure</strong> combines two or more basic shapes, break it into simpler pieces, then add or subtract as needed.</p>

<tr><th>Solid</th><th>Surface area formula</th></tr>

<tr><td>Rectangular prism</td><td>SA = 2(lw + lh + wh)</td></tr>

<tr><td>Cylinder</td><td>SA = 2πr² + 2πrh</td></tr>

<tr><td>Sphere</td><td>SA = 4πr²</td></tr>

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<p><strong>Worked Example:</strong> A cube has a surface area of 384 square inches. What is its volume?</p>

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<p class="prompt">1. A rectangular prism has length 4, width 3, and height 5. What is its surface area?</p>

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<p><span class="label">Explanation:</span> SA = 2[(4)(3) + (4)(5) + (3)(5)] = 2(12 + 20 + 15) = 2(47) = 94.</p>

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<p class="prompt">2. A cylinder has radius 4 and height 6. What is its total surface area, in terms of π?</p>

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<p><span class="label">Explanation:</span> SA = 2π(4)² + 2π(4)(6) = 32π + 48π = 80π.</p>

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<p class="prompt">3. A sphere has radius 5. What is its surface area, in terms of π?</p>

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<p><span class="label">Explanation:</span> SA = 4π(5)² = 100π.</p>

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<p class="prompt">4. A rectangular garden measures 15 feet by 8 feet. A walkway of uniform width 2 feet surrounds it. What is the total area of the garden and walkway combined?</p>

<button class="option-btn" data-correct="true" onclick="checkAnswer(this,'pd-4',true)">A) 228 square feet</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-4',false)">B) 120 square feet</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-4',false)">C) 174 square feet</button>

<button class="option-btn" data-correct="false" onclick="checkAnswer(this,'pd-4',false)">D) 342 square feet</button>

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<p><span class="label">Explanation:</span> The walkway adds 2 feet on every side, so the outer dimensions are 19 by 12: 19 × 12 = 228 square feet.</p>

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<p class="prompt">5. A cube has a surface area of 150 square inches. What is its volume?</p>

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<p><span class="label">Explanation:</span> 6s² = 150, so s² = 25 and s = 5. Volume: 5³ = 125.</p>

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

<li><strong>Answering with perimeter when the question asks for area, or vice versa.</strong> Underline exactly what's being asked before calculating, area uses square units, perimeter uses linear units.</li>

<li><strong>Forgetting to square the radius in a circle's area formula.</strong> A = πr², not πr, this is one of the single most common errors on circle problems.</li>

<li><strong>Confusing radius and diameter.</strong> Every formula involving a circle or sphere uses the radius. If a problem gives a diameter, divide by 2 before substituting.</li>

<li><strong>Forgetting the <span class="frac"><span class="num">1</span><span class="den">3</span></span> factor for cones and pyramids.</strong> A cone or pyramid has exactly one-third the volume of a cylinder or prism with the same base and height.</li>

<li><strong>Mixing up inconsistent units before calculating.</strong> A box listed as 2 feet by 18 inches by 1 foot needs every dimension in the same unit before multiplying, rewrite 18 inches as 1.5 feet first.</li>

</ul>

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

<h3>Which area and volume formulas are most important to know cold for the ACT?</h3>

<p>Rectangle area, triangle area, circle circumference and area, rectangular prism volume, and cylinder volume come up the most often. Sphere and cone volume are also worth knowing, since they're not easy to derive on the spot, even though the ACT occasionally provides less common formulas directly within a question.</p>

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<h3>How should I approach a composite figure problem?</h3>

<p>Break the figure into shapes you already have formulas for, calculate each piece separately, and then add or subtract them based on whether the pieces combine together or one is being removed from another.</p>

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<h3>Why does volume use cubic units while area uses square units?</h3>

<p>Area measures a two-dimensional region, so it scales with two multiplied dimensions (length × width), giving square units. Volume measures three-dimensional space, scaling with three multiplied dimensions, giving cubic units.</p>

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

<p>The <a href="https://theschoolofmathematics.com/quiz/act-perimeter-area-volume-quiz-1">Perimeter, Area & Volume 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-perimeter-area-volume-quiz-1">Practice Perimeter, Area & Volume 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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