What are the formulas for a rectangle?

Area = length × width (A = l × w). Perimeter = 2 × (length + width) (P = 2l + 2w). Diagonal = √(length² + width²) — from the Pythagorean theorem, since the diagonal forms a right triangle with the sides. Example: a rectangle 8m × 5m: area = 40 m², perimeter = 26 m, diagonal = √(64 + 25) = √89 ≈ 9.43 m. Properties that distinguish rectangles from other quadrilaterals: all four angles are 90°, opposite sides are equal and parallel, diagonals are equal in length and bisect each other (but don't bisect the angles unless it's a square). A square is a special rectangle where all sides are equal. For maximum area with a fixed perimeter: the optimal rectangle is a square. For P = 24m: a 6×6 square gives area = 36 m², while a 10×2 rectangle gives only 20 m². This principle (isoperimetric inequality) is why nature favors circular/spherical shapes for enclosing maximum volume — and why square rooms are more space-efficient than long narrow ones.

How do I find the dimensions of a rectangle given area and perimeter?

Given area (A) and perimeter (P), the dimensions are the solutions to the system: l × w = A and 2(l + w) = P. From the perimeter: l + w = P/2, so w = P/2 - l. Substitute into area: l × (P/2 - l) = A → l² - (P/2)l + A = 0. This is a quadratic in l. Solve with the quadratic formula: l = (P/2 ± √((P/2)² - 4A)) / 2. Example: area = 48, perimeter = 28. l + w = 14, l × w = 48. Quadratic: l² - 14l + 48 = 0. Discriminant: 196 - 192 = 4. l = (14 ± 2)/2 → l = 8 or l = 6. Dimensions: 8 × 6. The discriminant tells you if a solution exists: if (P/2)² - 4A 0, there are two valid dimensions. This problem appears in real life when you know fencing available (perimeter) and space needed (area), or when reverse-engineering dimensions from a floor plan.

What is the difference between a rectangle and a square?

A rectangle is a quadrilateral with four right angles and opposite sides of equal length, while a square is a special type of rectangle where all four sides are of equal length. For example, a rectangle with sides of 4 and 6 is not a square, but a rectangle with sides of 5 and 5 is a square. The area of a square can be calculated using the formula Area = side^2, which simplifies to 5^2 = 25 square units for a square with sides of 5 units.

How do I calculate the diagonal of a rectangle?

The diagonal of a rectangle can be calculated using the Pythagorean theorem, which states that the square of the length of the diagonal is equal to the sum of the squares of the lengths of the two sides. This can be expressed as diagonal^2 = length^2 + width^2. For example, the diagonal of a rectangle with a length of 8 and a width of 6 is calculated as diagonal^2 = 8^2 + 6^2, which simplifies to diagonal = sqrt(100) = 10 units.

What is the relationship between the area and perimeter of a rectangle?

The area of a rectangle is given by the formula Area = length * width, while the perimeter is given by the formula Perimeter = 2 * (length + width). For a rectangle with a length of 10 and a width of 5, the area is 10 * 5 = 50 square units, and the perimeter is 2 * (10 + 5) = 2 * 15 = 30 units. Understanding the relationship between area and perimeter is important in various applications, such as architecture and engineering.

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Variable Legend

l= lengthw= widthA= areaP= perimeterd= diagonal

Area

Space enclosed by the rectangle.

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Perimeter

Total distance around the rectangle.

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Diagonal

Length of a corner-to-corner line (Pythagoras).

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l= length

w= width

A= area

P= perimeter

d= diagonal

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