An unknown side from a perimeter
Perimeter is the total of all boundary sides. Subtract the known total, then divide the remainder among any equal unknown sides
\[P=K+x,\quad x=P-K\]
- Add every known side The known total K contains each given boundary length once \(\displaystyle K=6+7+8=21\text{ cm}\)
- Write the perimeter equation The total perimeter equals the known total plus the unknown side \(\displaystyle P=6+7+8+x\)
- Subtract the known total Rearrange the perimeter equation to isolate the unknown side \(\displaystyle x=32-21=11\text{ cm}\)
- Check the side length The answer must be positive and less than the whole perimeter \(\displaystyle 0<11<32\)
Worked example
A trapezium has perimeter 50 m. Three sides are 12 m, 16 m and 11 m. Find the fourth side
\[\begin{gathered}P=50\text{ m} \\ \text{Known: }12\text{ m},16\text{ m},11\text{ m}\end{gathered}\]
- \(\displaystyle K=12+16+11=39\text{ m}\)Add the three known boundary lengths
- \(\displaystyle 50=39+x\)The perimeter is the known total plus the fourth side
- \(\displaystyle x=50-39=11\text{ m}\)Subtract the known total to find the fourth side
Common mistakes
Adding instead of subtracting
✕\(\displaystyle x=27+16=43\text{ cm}\)✓\(\displaystyle x=27-16=11\text{ cm}\)
The known sides already form part of the perimeter, so their total must be subtractedConfusing perimeter and area
✕\(\displaystyle A=50=12+16+11+x\)✓\(\displaystyle P=50=12+16+11+x\)
The given total is a perimeter, which measures boundary length rather than areaMixing length units
✕\(\displaystyle \begin{gathered}x=3-80-70-50 \\ =-197\text{ cm}\end{gathered}\)✓\(\displaystyle \begin{gathered}3\text{ m}=300\text{ cm} \\ x=300-80-70-50 \\ =100\text{ cm}\end{gathered}\)
Convert all lengths to the same unit before subtracting themDifficulty levels
introFind a missing side in an isosceles triangle or trapezium from a whole-number perimeter
standardShare a regular polygon’s perimeter among its equal sides, including exact decimal lengths
challengeFind an unknown side in a mixed-unit or rectilinear boundary after converting units where needed
examRecognise an unknown side from three same-unit perimeter calculations
Questions about this topic
Why is the known total subtracted?+
The perimeter already includes the known sides, so subtracting them leaves the length of the missing side
Can a side found from a perimeter be negative?+
No. A side length must be positive, so a negative result shows that the equation or units need checking
What should I do when the sides use different units?+
Convert every length to one unit before adding or subtracting the boundary lengths
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 1: Measurement
In this chapter
Convert measurement unitsPerimeter of standard shapesFind an unknown side from a perimeterCircumference of a circlePerimeter of composite shapesArea of standard shapesArea of a circleArea of a sectorArea of composite shapes1F Prism surface area: Surface area of solidsSurface area of a triangular prismExposed surface area of joined cuboids1G Cylinder surface area: Surface area of solidsSurface area of a cylinder portionSurface area of a cylinder joined to a cuboidVolume of prisms and cylindersCapacity of a containerVolume of a cylinder portionJoined and hollow prism-cylinder volumesModel rainwater collection and storageSmallest unit and limits of accuracyMeasurement bounds in calculationsVolume of cones, pyramids and spheres (optional extension)1J Level 10A extension: pyramids, cones and spheres: Surface area of solids (optional extension)Surface areas of advanced solids and composites (optional extension)