Chapter 1 · Number Operations
1.1.4 Working Backwards
Area = side × side is a forward formula. Give yourself the area and ask for the side, and you're suddenly doing a square root — the exact same move, just read in reverse.
🎯 By the end of this you can…
- ✓Find the side of a square from its area, using a square root
- ✓Find the radius of a circle from its circumference, then its area
- ✓Find the side of a cube from its volume, using a cube root
💡 The idea
Every shape formula can be read backwards
If A=L2 gives you the area from the side, then L=A gives you the side back from the area. The same trick works for a cube: V=L3 forward, L=3V backward.
📐 See it
Given: Area of a square = 264 cm²
✏️ Worked example
Using that side, find the perimeter of the square
- 1Perimeter of a square =4L.
- 2Using the unrounded side (L≈16.24) before rounding the final answer: 4×16.24=64.96.
✅ Answer: ≈ 65.0 cm (3 s.f.)
📐 See it
Given: Circumference = 136 m (use π = 22/7)
✏️ Worked example
Using that radius, find the area of the circle
- 1Area =πr2=722×(21.6)2
- 221.62=466.56, so area =722×466.56=710264.32≈1466.3
✅ Answer: ≈ 1470 m² (3 s.f.)
📐 See it
Given: Area of a square garden = 331.24 m²
📐 See it
Given: Volume of a cube = 4913 cm³
✏️ Worked example
And the other direction: if one side of a cube is 12 m, find its volume
- 1V=L3=123
- 212×12=144, then 144×12=1728
✅ Answer: V=1728 m3
⚠️ The classic trap
It's tempting to square root a VOLUME or cube root an AREA out of habit. Match the operation to the dimension: area (two dimensions) reverses with a square root; volume (three dimensions) reverses with a cube root.
🎮 Work backwards to find the missing length
⭐ 0 XP🔥 streak 0🤔 A question gives you the VOLUME of a cube and asks for the side length. Which operation reverses that?
🏅
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