Perimeter: Formulas and Examples
Perimeter is the total length of a closed figure’s boundary. In brief: a rectangle with length 8 centimetres and width 3 centimetres has perimeter 8 + 3 + 8 + 3 = 22 centimetres. Perimeter measures distance around a figure, so it uses length units rather than square units.
The basic method is to follow the complete boundary and add every part once. Formulas shorten that process when a figure has known equal sides or repeated dimensions. This guide covers common polygons, missing lengths, composite outlines and the distinction between a shared internal edge and an exposed boundary.
What does perimeter measure?
Perimeter measures a one-dimensional path around a closed region. Imagine tracing the edge from a starting point until you return to that point. The total distance traveled along the boundary is the perimeter, provided each part is followed exactly once.
A polygon’s boundary consists of straight sides, so its perimeter is the sum of their lengths. A curved boundary needs an appropriate curve-length formula or measurement. The perimeter of a circle is usually called its circumference, while the same general idea of distance around the boundary applies.
Perimeter is different from area. A fence around a garden concerns perimeter, while the ground covered by the garden concerns area. Two figures can have the same perimeter and different areas, so knowing one measurement does not automatically determine the other.
How do you find the perimeter of any polygon?
Add all side lengths, using compatible units. A triangle with sides 5, 7 and 8 centimetres has perimeter 20 centimetres. A five-sided polygon with side lengths 2, 3, 4, 5 and 6 metres has perimeter 20 metres, even though its shape differs from the triangle.
The order of addition does not affect the total, but tracing the boundary in order helps prevent omissions or double counting. Mark each side as it is included, especially when the outline contains many short segments or turns inward.
Do not include diagonals or internal construction lines merely because their lengths are shown. A diagonal connects vertices through or across the figure and is not part of the outer polygon boundary. Identify the path being measured before choosing which labels to add.
What are the perimeter formulas for rectangles and squares?
A rectangle has two sides of length l and two of width w, so P = 2l + 2w = 2(l + w). For l = 8 centimetres and w = 3 centimetres, the formula gives 2(8 + 3) = 22 centimetres.
A square has four equal sides of length s, giving P = 4s. For a side of 4.5 metres, its perimeter is 18 metres. The square formula is also a special case of the rectangle formula with l = w = s.
Keep the formula connected to the boundary rather than treating it as an isolated symbol pattern. The factor 2 in the rectangle formula counts two copies of each dimension; the factor 4 for a square counts four equal sides. This explanation helps avoid confusing the formulas with area products.
| Figure | Required lengths | Perimeter |
|---|---|---|
| Triangle | Sides a, b, c | a + b + c |
| Rectangle | Length l, width w | 2(l + w) |
| Square | Side s | 4s |
| Parallelogram | Adjacent sides a, b | 2(a + b) |
| Regular n-gon | n sides, each length s | ns |
How do you find the perimeter of a regular polygon?
A regular polygon has equal side lengths, so multiply one side length by the number of sides. A regular hexagon with side 3.2 centimetres has perimeter 6 × 3.2 = 19.2 centimetres. There is no need to write six identical addends separately.
The same multiplication works for any polygon known to have all sides equal, even if its angles are not all equal. Regularity includes equal angles too, but equal side lengths alone are enough for the perimeter shortcut. Use only the conditions actually needed by the calculation.
A polygon’s name does not guarantee equal sides. An ordinary pentagon may have five different side lengths. Unless regularity or equal-side marks are given, calculate its perimeter from all the side data rather than multiplying one arbitrary side by five.
How do you find a missing side from the perimeter?
Subtract the total known boundary length from the perimeter. If a quadrilateral has perimeter 29 centimetres and three sides measure 6, 8 and 7 centimetres, the missing side is 29 − (6 + 8 + 7) = 8 centimetres.
For a rectangle, repeated sides make an equation useful. If P = 30 metres and l = 9 metres, then 2(9 + w) = 30. Divide by 2 to get 9 + w = 15, then subtract 9 to obtain w = 6 metres.
Check the recovered length in the full boundary sum. For the rectangle, 9 + 6 + 9 + 6 = 30. Also check that the length is positive and consistent with any geometric conditions; a negative answer usually indicates inconsistent data or an incorrect equation.
- Use the rectangle formula 2(l + w) = P.
- Substitute l = 9 m and P = 30 m.
- Solve 9 + w = 15 to obtain w = 6 m.
- Check 2(9 + 6) = 30 m.
How do you find the perimeter of a composite figure?
Trace only the exposed boundary of the combined region. If two rectangles share an edge, that shared edge is internal and does not belong to the outside perimeter. Adding both rectangle perimeters counts the shared edge twice, so subtract twice its length when the joining arrangement is otherwise nonoverlapping.
For rectangles with perimeters 20 centimetres and 14 centimetres sharing a complete 3-centimetre boundary segment, the combined perimeter is 20 + 14 − 2 × 3 = 28 centimetres. The assumption about the shared segment is essential; a different overlap can create a different boundary.
For a rectilinear outline with missing lengths, use the total horizontal and vertical spans to infer them where the layout supports that relation. Do not automatically use the bounding rectangle’s perimeter: inward notches can add boundary length. A corner cutout may replace two removed segments with equal-length new segments, while an interior notch can behave differently.
How do units and practical allowances affect a perimeter problem?
Convert all lengths to one unit before adding. A rectangle measuring 2 metres by 75 centimetres has dimensions 200 centimetres by 75 centimetres, so its perimeter is 550 centimetres or 5.5 metres. Adding 2 and 75 without conversion would mix incompatible unit counts.
A practical material question may differ from the bare mathematical perimeter. Fencing can exclude a gate opening, while ribbon or trim may require extra length for joins. Apply only the allowances stated in the problem rather than adding an arbitrary percentage.
For example, a boundary of 24 metres with a 1.5-metre opening requires 22.5 metres of continuous fencing before any specified join allowance. The subtraction follows from the portion actually covered. The perimeter remains 24 metres even though the material length is smaller because part of the boundary is left open.
Why do perimeter formulas work and which mistakes are common?
Perimeter formulas are compressed versions of boundary addition. Their background is length measurement along a path, with geometric properties identifying repeated side lengths. A rectangle, square or regular polygon permits a shortcut because its side relationships reduce the number of separate measurements needed.
Common mistakes include multiplying rectangle dimensions instead of adding them, using square units for perimeter and including internal diagonals. Another is assuming all sides are equal because the sketch looks regular. A valid formula must match the figure’s stated properties.
Check by tracing the boundary aloud or labeling each segment in order, then compare the formula result with an explicit sum. Use a length unit in the final answer and distinguish the mathematical boundary from any material allowance. These habits make perimeter problems manageable even when the outline is irregular.