Rectangle and Square
A rectangle is a quadrilateral with four right angles, and a square is a rectangle with four equal sides. In brief: a rectangle with length l and width w has area lw, perimeter 2(l + w) and diagonal √(l2 + w2). A square with side s has area s2, perimeter 4s and diagonal s√2. For a 6 by 8 rectangle the diagonal is 10.
Rectangles and squares are the most common shapes in everyday measurement, and their formulas are the basis for areas of other figures. This guide explains the definitions and properties, derives the formulas for area, perimeter and diagonals, shows how to solve typical problems and compares the two shapes in a table.
What are the properties of a rectangle?
- All four angles are 90°.
- Opposite sides are parallel and equal, so it is a parallelogram.
- The diagonals are equal and bisect each other, so their intersection point is the center of symmetry and the center of the circumscribed circle.
- It has two axes of symmetry (the lines through the midpoints of opposite sides) and rotational symmetry of order 2.
The converse tests: a parallelogram with one right angle is a rectangle, and a parallelogram with equal diagonals is a rectangle. A quadrilateral with three right angles is a rectangle, since the fourth angle is 360° − 270° = 90°.
What are the properties of a square?
- All sides are equal and all angles are 90°.
- The diagonals are equal, bisect each other, are perpendicular and bisect the angles, so each makes 45° with the sides.
- It has four axes of symmetry (two diagonals, two midlines) and rotational symmetry of order 4.
- It is a rectangle, a rhombus and a parallelogram at the same time.
What are the formulas?
Rectangle (length l, width w):
- Area A = l · w
- Perimeter P = 2l + 2w
- Diagonal d = √(l2 + w2), by the Pythagorean theorem
Square (side s):
- Area A = s2 = d2/2
- Perimeter P = 4s
- Diagonal d = s√2 ≈ 1.414s
- Circumradius R = s√2/2 and inradius r = s/2
How do you solve typical problems?
Example 1: diagonal
A rectangle has sides 9 cm and 12 cm. Diagonal: √(81 + 144) = √225 = 15 cm. Area: 108 cm2. Perimeter: 42 cm.
Example 2: a missing side
A rectangle has perimeter 34 m and area 60 m2. Then l + w = 17 and lw = 60. The numbers are roots of t2 − 17t + 60 = 0, so t = 12 and t = 5 (by Vieta’s formulas, or by factoring (t − 12)(t − 5)). The sides are 12 m and 5 m, and the diagonal is 13 m.
Example 3: a square from the diagonal
A square has diagonal 10 cm. Then s = 10/√2 = 5√2 ≈ 7.07 cm, and the area is d2/2 = 50 cm2.
Example 4: the maximum area
Among all rectangles with perimeter 40, the area l(20 − l) is largest at l = 10, where the rectangle is a square with area 100. A rectangle 8 by 12 has area 96, 5 by 15 has 75. This follows from the vertex of a parabola.
Example 5: a square and a rectangle with equal perimeters
A square and a rectangle 3 cm longer than wide have the same perimeter 36 cm. The square has side 9 and area 81. The rectangle has w + (w + 3) = 18, so w = 7.5 and l = 10.5, with area 78.75. The square has the larger area.
Rectangle and square compared
| Quantity | Rectangle l × w | Square s × s |
|---|---|---|
| Area | lw | s2 |
| Perimeter | 2(l + w) | 4s |
| Diagonal | √(l2 + w2) | s√2 |
| Axes of symmetry | 2 | 4 |
| Diagonals perpendicular | only if l = w | always |
| Circumradius | d/2 | s√2/2 |
| Incircle | none (unless a square) | radius s/2 |
Where do these shapes come from?
Right angles and rectangles were fundamental in ancient surveying and construction. The Egyptian and Babylonian scribes calculated the areas of rectangular fields, and Euclid defined the square and the oblong (rectangle) in Book I of the Elements. The word “square” for the second power of a number comes from the area of a square: Greek mathematicians described x2 as a geometric square on a segment of length x.
Where are they used today?
Rooms, screens, paper, bricks and city blocks are rectangles. Flooring and painting calculations rely on the area formula: a room 4.5 m by 3.2 m has an area of 14.4 m2. Screen sizes are given as the diagonal: a 16:9 screen with a 27-inch diagonal has width 27 · 16/√337 ≈ 23.5 inches and height ≈ 13.2 inches, because √(162 + 92) = √337.
Which mistakes are common?
Mixing up perimeter and area
Perimeter has units of length (cm) and area has square units (cm2). Doubling the sides doubles the perimeter but quadruples the area.
Assuming that the diagonal is l + w
The diagonal is √(l2 + w2), which is shorter than l + w.
Assuming a rectangle has perpendicular diagonals
Only the square (and the rhombus) has perpendicular diagonals.
Calling a rectangle with equal sides by the wrong name
A rectangle with all sides equal is a square. Both names are correct, but “square” is more specific.