Geometry & Shapes
Find Rectangle Dimensions from Area and Perimeter
Recover rectangle sides from area and perimeter, solve the resulting quadratic, and identify impossible or insufficient measurement combinations.
Use the two independent equations
If a rectangle has area A and perimeter P, its sides satisfy lw = A and l + w = P/2. Substitute w = P/2 − l into the area equation. This gives l² − (P/2)l + A = 0. Solving this quadratic gives the two side lengths, with the roots interchangeable as length and width. Supplying area and perimeter is different from supplying only one of them; each equation contributes information needed to identify the pair.
Work a pair with integer dimensions
Suppose area is 40 cm² and perimeter is 26 cm. Then l + w = 13 and lw = 40. The quadratic l² − 13l + 40 = 0 factors as (l − 5)(l − 8) = 0. The dimensions are therefore 5 cm and 8 cm. Check by multiplication, 5 × 8 = 40, and by boundary sum, 2(5 + 8) = 26. Entering those dimensions in the rectangle tool should reproduce both supplied measurements.
Detect an impossible combination
The quadratic discriminant is (P/2)² − 4A. It must be nonnegative for real side lengths. With perimeter 20 and area 30, the discriminant is 100 − 120 = −20, so no real rectangle fits both values. A square with that perimeter has sides 5 and area 25, the largest possible rectangular area for that perimeter. An area of 30 therefore contradicts the perimeter rather than requiring a calculator to invent complex-valued physical dimensions.
Use a ratio or diagonal when that is the extra condition
If length is three times width and area is 75, then 3w² = 75, giving w = 5 and l = 15. If diagonal is 13 and one side is 5, the other is √(169 − 25) = 12. These are separate information patterns and should not be mixed without checking consistency. A room measurement may be approximate, so two supplied quantities can disagree slightly; preserve the measured values and explain the approximation instead of silently altering one to force an exact rectangle. When the two quadratic roots are equal, the rectangle is a square; that repeated-root case is valid and marks the largest area at the stated 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
Solve for length and width using the relationships below, then enter the resulting pair to check both area and perimeter. The tool’s two fields expect dimensions, not area and perimeter.
Sources and method
Related guides
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