Inside and Outside: Volume and Surface Area

Imagine you have a cardboard box. You need to do two separate jobs: fill it with popcorn, and wrap it with gift paper. Would you need the same measurement for both jobs? Think about it — filling the inside and covering the outside are completely different challenges. Today you are going to find the maths that handles each one. Before we start, recall this: the area of a rectangle is length × width. How would you find the area of a rectangle that is 4 cm long and 3 cm wide?

A cuboid is a 3-D box shape where every face is a rectangle — like a brick, a book, or a cereal box. It has three key measurements: length (l), width (w), and height (h). **Volume** measures how much space is inside a 3-D shape. Think of it as how many unit cubes (little 1 cm × 1 cm × 1 cm cubes) you could pack inside. Volume is measured in cubic units, written cm³ (read 'cubic centimetres') or m³. Formula: V = l × w × h Multiplying the three dimensions together tells you how many unit cubes fit — one layer of (l × w) cubes, stacked h layers high. **Surface area** measures the total area of all the faces on the outside of a shape. Think of it as how much wrapping paper you would need to cover every face completely. Surface area is measured in square units, written cm² (read 'square centimetres'). A cuboid has 6 faces arranged in 3 pairs of identical rectangles: • Top and bottom — each has area l × w, so together: 2(l × w) • Front and back — each has area l × h, so together: 2(l × h) • Left and right sides — each has area w × h, so together: 2(w × h) Formula: SA = 2(lw) + 2(lh) + 2(wh) You can also write this as SA = 2(lw + lh + wh). **Worked example** A cuboid has length 5 cm, width 3 cm, and height 2 cm. Volume: V = 5 × 3 × 2 = 30 cm³ Surface area: • Top and bottom: 2 × (5 × 3) = 2 × 15 = 30 cm² • Front and back: 2 × (5 × 2) = 2 × 10 = 20 cm² • Left and right: 2 × (3 × 2) = 2 × 6 = 12 cm² SA = 30 + 20 + 12 = 62 cm² So the box holds 30 cm³ of space inside, and you would need 62 cm² of wrapping paper to cover it.