Perimeter Area & Volume

<p>Measuring the outside, the surface, and the space inside shapes are three core geometry skills. This guide covers the formulas you need and shows you how to apply them step by step.</p>

<h2>Perimeter</h2>

<p>The <strong>perimeter</strong> is the total distance around the outside of a 2D shape. Add up all the side lengths.</p>

<p><strong>Worked Example:</strong> A rectangle is 12 cm long and 5 cm wide. Perimeter = 12 + 5 + 12 + 5 = 34 cm. Or use the shortcut: P = 2(l + w) = 2(12 + 5) = 34 cm.</p>

<p>For irregular shapes, make sure you find every side length. If some sides are not labelled, work them out from the other given measurements.</p>

<h2>Area of Rectangles and Squares</h2>

<p>Area = length × width. A rectangle measuring 8 cm by 6 cm has area = 8 × 6 = 48 cm². Always write the unit as cm² ("square centimetres").</p>

<h2>Area of a Triangle</h2>

<p>Area = base × height ÷ 2. The height must be measured at right angles (perpendicular) to the base.</p>

<p><strong>Worked Example:</strong> A triangle has a base of 10 cm and a height of 7 cm. Area = 10 × 7 ÷ 2 = 35 cm².</p>

<h2>Area of a Parallelogram</h2>

<p>Area = base × perpendicular height. Note: do NOT use the slanted side — use the vertical height.</p>

<h2>Area of a Trapezium</h2>

<p>Area = (a + b) ÷ 2 × h, where a and b are the two parallel sides and h is the perpendicular height between them.</p>

<p><strong>Worked Example:</strong> A trapezium has parallel sides of 6 cm and 10 cm and a height of 4 cm. Area = (6 + 10) ÷ 2 × 4 = 8 × 4 = 32 cm².</p>

<h2>Compound Shapes</h2>

<p>Split the shape into rectangles (or triangles) you know how to handle, find each area, and add them up.</p>

<p><strong>Worked Example:</strong> An L-shape can be split into two rectangles. Rectangle 1: 8 × 3 = 24 cm². Rectangle 2: 4 × 5 = 20 cm². Total area = 44 cm².</p>

<h2>Volume of Cuboids</h2>

<p>Volume = length × width × height. The unit is cm³ ("cubic centimetres").</p>

<p><strong>Worked Example:</strong> A cuboid is 5 cm long, 4 cm wide, and 3 cm tall. Volume = 5 × 4 × 3 = 60 cm³.</p>

<h2>Volume of Other Prisms</h2>

<p>Volume = area of cross-section × length. A triangular prism with a triangular face of area 12 cm² and a length of 10 cm has volume = 12 × 10 = 120 cm³.</p>

<h2>Surface Area</h2>

<p>The surface area of a 3D shape is the total area of all its faces. For a cuboid: SA = 2(lw + lh + wh).</p>

<p><strong>Worked Example:</strong> Cuboid 5 × 4 × 3. SA = 2(20 + 15 + 12) = 2 × 47 = 94 cm².</p>

<h2>Units</h2>

<ul>

<li>Perimeter → cm, m, mm (length units)</li>

<li>Area → cm², m², mm² (squared units)</li>

<li>Volume → cm³, m³, mm³ (cubed units)</li>

</ul>

<h2>Practice Tips</h2>

<ul>

<li>Always check which measurement is the HEIGHT (perpendicular) — not the slanted side</li>

<li>For compound shapes, draw lines to show how you split them and label every dimension</li>

<li>Write the correct unit with every answer — marks are often lost here</li>

<li>Estimate first: a rectangle 8 × 6 should be a bit less than 8 × 8 = 64; your answer of 48 makes sense</li>

</ul>