How do you find the area and perimeter of a rectangle?

The area of a rectangle is A = l × w (length times width). The perimeter is P = 2(l + w) — the total distance around all four sides. Both formulas require only the two side lengths. For a rectangle with length 8 and width 5: • Area = 8 × 5 = 40 square units • Perimeter = 2(8 + 5) = 2 × 13 = 26 units • Diagonal = √(8² + 5²) = √(64 + 25) = √89 ≈ 9.434 units Remember: • Area is in square units (cm², m², ft²) — always squared • Perimeter is in the same units as the sides (cm, m, ft) — not squared • A square is a special rectangle where l = w, so A = l² and P = 4l • This calculator solves from any two known measurements — not just length and width

How do you find the diagonal of a rectangle?

The diagonal of a rectangle is found using the Pythagorean theorem, because the diagonal divides the rectangle into two right triangles. If length = l and width = w, then diagonal d = √(l² + w²). The diagonal is always longer than either side but shorter than l + w. Examples: • l=8, w=5: d = √(64+25) = √89 ≈ 9.434 • l=6, w=6 (square): d = √(36+36) = 6√2 ≈ 8.485 • l=3, w=4: d = √(9+16) = √25 = 5 (this is the 3-4-5 right triangle!) Inverse — finding a side from diagonal: • Given d and l: w = √(d²−l²) • Given d and w: l = √(d²−w²) • Both diagonals of a rectangle are equal in length and bisect each other • The diagonals of a square bisect each other at right angles (90°)

How do you solve for length and width given area and perimeter?

If you know both area (A) and perimeter (P), you can find length and width by solving a quadratic equation. From P = 2(l+w), we get l+w = P/2. Combined with l×w = A, we have a system: the sum and product of two numbers. This leads to the quadratic x² − (P/2)x + A = 0. Step-by-step for A=40, P=26: • l + w = 26/2 = 13 • l × w = 40 • Quadratic: x² − 13x + 40 = 0 • Discriminant: 13² − 4×40 = 169 − 160 = 9 • l = (13 + √9)/2 = (13+3)/2 = 8 • w = (13 − √9)/2 = (13−3)/2 = 5 • Check: 8+5=13 ✓, 8×5=40 ✓ Note: if discriminant < 0, no real rectangle exists with those values.

What is the golden rectangle?

A golden rectangle has length-to-width ratio equal to the golden ratio φ (phi) = (1+√5)/2 ≈ 1.61803. This ratio appears extensively in art, architecture, and nature. A golden rectangle has the special property that when you remove a square from one end, the remaining rectangle is also a golden rectangle — you can repeat this process infinitely. Properties of the golden rectangle: • l/w = φ ≈ 1.61803 • If width = 1: length = φ ≈ 1.618 • Removing a square leaves a smaller golden rectangle • The spiral drawn through these squares approximates the Fibonacci spiral Historical appearances: • Parthenon in Athens (ancient Greece) • Leonardo da Vinci's Vitruvian Man • Many credit card and smartphone aspect ratios • Nautilus shell spirals approximate the golden spiral • This calculator identifies golden rectangles and highlights them with a badge

What is the difference between a rectangle and a square?

A square is a special type of rectangle where all four sides are equal (l = w). Every square is a rectangle, but not every rectangle is a square. Both have four right angles (90°). The key difference is that a rectangle has two pairs of equal sides while a square has all four sides equal. Key differences: • Rectangle: l ≠ w (usually), four right angles, two pairs of equal parallel sides • Square: l = w, four right angles, all four sides equal • Rectangle area: A = l × w • Square area: A = l² (since l = w) • Rectangle diagonal: d = √(l²+w²) • Square diagonal: d = l√2 (since both sides equal l) • Rectangle perimeter: P = 2(l+w) • Square perimeter: P = 4l • A square has additional symmetry: diagonals are equal AND perpendicular AND bisect the corner angles (45° each) • This calculator detects when l = w and labels the result as a square

Rectangle Calculator — Area, Perimeter, Diagonal & More

The Rectangle Calculator solves all rectangle properties from any two known measurements. Seven input modes: length + width (most common), length + area, length + perimeter, length + diagonal, width + area, width + diagonal, or area + perimeter (solved using a quadratic). All results are shown insta...

RECTANGLE CALCULATOR

KNOWN VALUES

Length (l)

Width (w)

LENGTH (l)

WIDTH (w)

AREA

40 units²

PERIMETER

26 units

DIAGONAL

9.433981 units

RATIO l:w

1.6

Half perimeter13

Area/perimeter1.5385

Diagonal angle32.01°

TypeRectangle

STEP-BY-STEP

Given: length = 8, width = 5

Area = l × w = 8 × 5 = 40

Perimeter = 2(l+w) = 2(8+5) = 26

Diagonal = √(l²+w²) = √(64+25) = 9.433981

QUICK EXAMPLES

LIVE DIAGRAM

DIAGONAL (PYTHAGORAS)

d = √(l²+w²)
= √(8²+5²)
= √(64+25)
= √89 = 9.433981

DIMENSIONS

Length 8100%

Width 562.5%

Diagonal 9.434117.9%

Created with❤️byeaglecalculator.com

RECTANGLE FORMULAS

ALL FORMULAS

NAME	FORMULA	DESCRIPTION
Area	A = l × w	Length times width
Perimeter	P = 2(l + w)	Twice the sum of length and width
Diagonal	d = √(l² + w²)	Pythagorean theorem on the right triangle formed
Width from area	w = A / l	Divide area by known length
Width from diagonal	w = √(d² − l²)	Rearrange Pythagorean theorem
From area+perimeter	x² − (P/2)x + A = 0	Quadratic equation to find both sides
Square (special case)	l = w → d = l√2, A = l², P = 4l	All sides equal
Golden rectangle	l/w = φ = (1+√5)/2 ≈ 1.61803	Aesthetically proportioned rectangle

HOW TO USE

1
Select your input mode based on which two values you know. The most common is Length + Width, but any two of the five properties (l, w, A, P, d) can be used to find the rest.
2
Enter your two known values. For the Area + Perimeter mode, the calculator solves a quadratic equation to find both sides simultaneously.
3
Area, perimeter, diagonal, and the l:w ratio appear instantly. The type (Rectangle or Square) is shown, along with a Golden Rectangle badge if l/w ≈ 1.618.
4
Read the step-by-step solution to see the formula applied — Pythagorean theorem for diagonal problems, quadratic for Area + Perimeter mode.
5
Check the live diagram — the rectangle is drawn to scale with the diagonal in red. A Pythagoras proof panel shows the diagonal calculation step by step.

WORKED EXAMPLE

Example 1 (l+w): l=8, w=5. Area=40, Perimeter=26, Diagonal=√89≈9.434. Example 2 (square l=6): Area=36, Perimeter=24, Diagonal=6√2≈8.485. Type=Square. Example 3 (A+P=40,26): l+w=13, l×w=40, quadratic→l=8,w=5. Example 4 (l+diagonal l=8,d=√89): w=√(89−64)=√25=5.

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Last updated: April 29, 2026 · Formula verified by EagleCalculator team · Eagle-eyed accuracy for every calculation.