<h3>The Main Idea</h3> <p><strong>Volume is how much space a 3D shape takes up.</strong></p> <p>Think of it as "how much can fit inside" — measured in cubic units (ft³, in³, cm³).</p> <h3>Rectangular Prism (Box)</h3> <p>Multiply all three dimensions:</p> $$V = l \times w \times h$$ <p>Length × Width × Height — the order doesn't matter.</p> <h3>Cube</h3> <p>A cube is a special box where all sides are equal:</p> $$V = s^3 = s \times s \times s$$ <p>Same number multiplied three times.</p> <h3>Cylinder</h3> <p>Area of the circular base times the height:</p> $$V = \pi r^2 h$$ <p>Find the circle area first (πr²), then multiply by height.</p> <h3>Working Backwards</h3> <p>If you know the volume and some dimensions, divide to find the missing one:</p> <ul> <li>For a box: missing dimension = V ÷ (other two dimensions)</li> <li>For a cylinder: if finding h, use h = V ÷ πr²</li> <li>For a cylinder: if finding r, use r² = V ÷ πh, then take the square root</li> </ul> <h3>Watch the Units!</h3> <p>All dimensions must be in the <strong>same unit</strong> before multiplying:</p> <ul> <li>1 yard = 3 feet</li> <li>1 foot = 12 inches</li> <li>27 cubic feet = 1 cubic yard (3 × 3 × 3)</li> </ul>
Practice This Goal<h3>The Main Idea</h3> <p><strong>Surface area is the total area of all the faces of a 3D shape.</strong></p> <p>When a problem asks how much material is needed to cover, wrap, or paint the outside of an object, you're finding surface area. The key giveaway is "square" units (square inches, square feet) combined with references to faces, sides, or the outside of a 3D object.</p> <h3>How to Identify Surface Area Problems</h3> <p>Look for these clues:</p> <ul> <li><strong>Square units</strong> — "square inches," "square feet" (NOT cubic units, which indicate volume)</li> <li><strong>References to faces or sides</strong> — "all faces," "every side," "the outside"</li> <li><strong>Real-world covering</strong> — wrapping, painting, covering with tape, tiling</li> <li><strong>Three dimensions given</strong> — length, width, AND height (or radius and height for cylinders)</li> </ul> <h3>Working with π</h3> <p>For cylinder problems:</p> <ul> <li><strong>Leave answers in terms of π</strong> unless told otherwise (e.g., "72π square inches")</li> <li><strong>Use π = 3.14</strong> only when the problem specifically instructs you to</li> </ul> <p>Keeping π as π gives exact answers and avoids rounding errors.</p> <h3>The Approach</h3> <ol> <li><strong>Read the question sentence first</strong> — Know what you're solving for before reading the details.</li> <li><strong>Identify the shape</strong> — Is it a rectangular box (prism), a cube, or a cylinder?</li> <li><strong>Write the correct formula</strong> — Each shape has its own surface area formula.</li> <li><strong>Plug in what you know</strong> — Substitute the given values.</li> <li><strong>Solve</strong> — Either calculate directly, or solve the equation if working backwards.</li> </ol> <h3>Working Backwards</h3> <p>If you're given the surface area and asked to find a missing dimension:</p> <ul> <li>Plug in all known values into the formula</li> <li>Simplify what you can</li> <li>Solve the resulting equation for the unknown</li> </ul> <p>This requires comfort with solving equations — if that's tricky, practice that skill separately.</p>
Practice This Goal