Perimeter of polygons
Trace the complete outside boundary and add its lengths. Equal-side properties reduce repeated addition
- Trace the boundary Add each outside side exactly once. Exclude heights and diagonals that lie inside the shape. For a triangle with sides a, b and c: \(\displaystyle P=a+b+c\)
- Use equal sides A square or rhombus has four equal sides. A rectangle or parallelogram has two pairs of equal opposite sides \(\displaystyle \begin{gathered}P=4s\\P=2a+2b\end{gathered}\)
- Match units first Convert all boundary lengths to the same unit before adding. State a length unit, not a square unit \(\displaystyle 250\text{ cm}=2.5\text{ m}\)
Worked example
A parallelogram has base 9.2 m, sloping side 3.5 m and perpendicular height 2.8 m. Find its perimeter
\[\begin{gathered}b=9.2\text{ m}\\s=3.5\text{ m}\\h=2.8\text{ m}\end{gathered}\]
- \(\displaystyle P=2b+2s\)Opposite sides are equal; the internal height is not a boundary side
- \(\displaystyle P=2(9.2)+2(3.5)\)Substitute the base and sloping-side lengths
- \(\displaystyle P=25.4\text{ m}\)State the complete boundary length in metres
Common mistakes
Adding an internal height
✕\(\displaystyle P=7.5+10+12.5+6\)✓\(\displaystyle P=7.5+10+12.5=30\text{ m}\)
The height of 6 m is inside this triangle. Count only the three boundary sidesCounting only half the boundary
✕\(\displaystyle P=6+4=10\text{ cm}\)✓\(\displaystyle P=2(6)+2(4)=20\text{ cm}\)
A rectangle has two sides of each lengthMixing units
✕\(\displaystyle P=2(250)+2(1.6)\)✓\(\displaystyle \begin{gathered}250\text{ cm}=2.5\text{ m}\\P=2(2.5)+2(1.6)\\P=8.2\text{ m}\end{gathered}\)
Convert all sides to the same unit before addingDifficulty levels
introFind square and rectangle perimeters
standardFind triangle, parallelogram and rhombus perimeters, distinguishing boundary sides from internal dimensions
Questions about this topic
Does a dashed height count towards perimeter?+
Only if it is actually a boundary side. A height drawn inside the shape is not part of the perimeter
Do I need the diagonals of a rhombus?+
Not when its side length is already given. Multiply that side length by four
Why do we use ordinary units?+
Perimeter measures a length around the shape. Area measures the enclosed region and uses square units
Where do I practise finding a missing side?+
Use the missing-side topic in this same section. It starts with a known perimeter and works backwards
Can I use a calculator?+
A calculator is allowed unless a question is marked no 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)