Geometry & Shapes
Perimeter Formulas: Rectangles, Polygons and Circles
Choose a perimeter formula by shape, distinguish circle circumference, and add compound boundaries without including shared internal edges.
Begin with the outside boundary
Perimeter is the total length of a closed figure’s boundary. For a polygon, the general rule is to add every side. Special formulas shorten that addition when the shape has known equalities. A rectangle gives P = 2(l + w), a square gives P = 4s, and an equilateral triangle gives P = 3s. The last shortcut requires all three sides to be equal; a general triangle instead needs P = a + b + c.
Choose the correct circle quantity
A circle’s boundary length is called circumference, C = 2πr = πd. Radius r and diameter d are not interchangeable fields. For radius 3 cm, circumference is 6π cm, approximately 18.8496 cm. The area 9π cm² is a different output. For a semicircular region, its entire perimeter includes both the curved half-circumference and the straight diameter: P = πr + 2r. Using only πr counts the arc and leaves out part of the closed boundary.
Apply equal-side multiplication carefully
For a regular octagon with side 2.5 m, perimeter is 8 × 2.5 = 20 m. An irregular octagon with the same first side needs its other seven lengths as well. A known perimeter of 20 for a regular octagon can be inverted to side 20/8 = 2.5. The side count is a positive integer at least three; a fractional count is not a valid polygon input. Equal angles are not needed for the multiplication if equal side lengths are independently known.
Remove shared edges from compound sums
If two rectangles are joined along an edge, adding their separate perimeters counts the shared edge twice even though it is internal to the combined region. For rectangles 6 by 4 and 3 by 4 joined along their 4-unit sides, separate perimeters total 20 + 14 = 34. Subtract twice the shared length, 8, to get 26, matching the perimeter of the combined 9-by-4 rectangle. For a more complicated arrangement, trace the actual exposed boundary instead of relying on an unverified joining shortcut. Only exposed edges contribute to the combined outside perimeter.
Separate boundary length from covered area
Perimeter measures the length around a boundary. Area measures the amount of flat surface enclosed by that boundary. A length is expressed in units such as metres, whereas an area is expressed in square metres. The two quantities cannot be exchanged by relabelling the answer. A narrow rectangle and a nearly square rectangle can have the same perimeter but different areas. Decide whether the task concerns an edge, a covering, or a missing dimension before selecting the formula or interpreting a calculator’s first displayed result.
Convert dimensions before combining them
Use one length unit for all dimensions in a formula. A rectangle described as 2 metres by 50 centimetres can be calculated as 2 × 0.5 = 1 square metre or 200 × 50 = 10,000 square centimetres. Both describe the same area. Since 1 metre equals 100 centimetres, 1 square metre equals 100 × 100 = 10,000 square centimetres. Apply the conversion to both dimensions, or square the linear conversion factor when converting an already calculated area. A single factor of 100 would be incorrect for that area conversion.
Check dimensions and the shape assumptions
A formula applies to the stated ideal shape and the measurements it actually requires. A rectangle needs perpendicular adjacent sides; a parallelogram’s height is perpendicular to its base; a triangle’s height is not automatically one of its sloping sides. Draw and label the figure before entering numbers. Positive dimensions, consistent units and the appropriate geometric relationships should all remain true after solving for a missing value. Retain exact expressions through the working when possible and round only the final result to the requested precision or the precision supported by the measurements.
Calculator for the supported step
Choose the shape in the Perimeter Calculator, then fill only the dimensions requested for that shape. Its polygon mode assumes equal sides and uses side count times side length.
Sources and method
Related guides
Form a perimeter equation from algebraic side labels, solve a rectangle example, and verify that the resulting sides are positive and consistent.
Add algebraic side expressions to find perimeter, combine like terms, and use supplied values without assuming every variable must be solved.
Distinguish a missing total perimeter from a missing side, use shape relationships, and calculate an L-shaped boundary without counting internal lines.