- A = Length x Breadth
- Perimeter formula: P = 2 x (Length + Breadth)
- The diagonal D satisfies D2 = L2 + B2 (Pythagorean theorem)
- All angles in a rectangle are 90 degrees, and opposite sides are equal and parallel.
A rectangle can be considered both a quadrilateral and a parallelogram since it contains two pairs of lines that are parallel to each other. Opposite sides are of equal lengths, and both pairs are of different lengths.
A rectangle can also be considered an equiangular quadrilateral since each side meets each other at an angle of 90 degrees.
Whether or not a quadrilateral is a rectangle can be determined if any of the following conditions are true:
- A parallelogram that has at least one right angle.
- A parallelogram whose diagonals are of equal length.
- Triangles ABD and DCA of a parallelogram ABCD are congruent.
- A quadrilateral that has four right angles.
- An equiangular quadrilateral.
What Are the Properties of a Rectangle?
A rectangle is a quadrilateral with four right angles of 90 degrees each. It has two pairs of parallel sides of equal length, two equal diagonals that bisect each other, and is both cyclic and equiangular. A rectangle is also a special case of a parallelogram with all angles equal to 90 degrees.
Symmetrical Properties of a Rectangle
- Cyclic: All corners of a rectangle lie on a circle. A rectangle is a cyclic polygon.
- Equiangular: All angles of a rectangle are equal (each 90 degrees). A rectangle is an equiangular polygon.
- Isogonal: All angles lie in a symmetrical orbit. A rectangle is an isogonal polygon.
- Rectilinear: Its sides meet at right angles. A rectangle is a rectilinear polygon.
- Circumcircle: The centre is equidistant from all vertices. A rectangle has a circumscribed circle.
What Theorems Apply to Rectangles?
The isoperimetric theorem states that among all rectangles with a given area, the square has the smallest perimeter. A crossed rectangle has two opposite sides of a standard rectangle and two diagonals, forming a butterfly or bowtie shape. Rectangles also satisfy the Pythagorean theorem: D2 = L2 + B2.
- Isoperimetric theorem: A square has the largest area amongst all rectangles of a given perimeter.
- Crossed rectangle: Has two opposite sides of a standard rectangle and two diagonals, forming a butterfly or bowtie shape.
- Pythagorean theorem: The diagonal D of a rectangle satisfies D2 = Length2 + Breadth2.
What Is the Geometry of a Rectangle?
Every rectangle has four sides, four right angles of 90 degrees each, and two diagonals of equal length. The sides are labelled AB, BC, CD, and DA. The diagonals are AC and BD. All four angles A, B, C, and D are right angles, and opposite sides are equal and parallel.
- Sides: The sides of a rectangle are AB, BC, CD, and DA.
- Diagonals: The diagonals are AC and BD, and both are of equal length.
- Angles: All four angles A, B, C, and D are right angles measuring 90 degrees.
How Do You Calculate the Perimeter of a Rectangle?
The perimeter of a rectangle is the total length of all four sides. Since opposite sides are equal, the perimeter formula is P = 2 x (length + breadth). For example, if a rectangle has sides of length A and breadth B, its perimeter is 2A + 2B = 2(A + B). All measurements must be in the same unit.
The perimeter of a rectangle is the sum of all four sides. The formula is:
Perimeter = 2 x (Length + Breadth)
For example, if the lengths of the sides of a rectangle are A and B, then the perimeter is:
Perimeter = 2A + 2B = 2(A + B)
Area of an Equilateral Triangle: Formula and ExamplesRead →How Do You Calculate the Area of a Rectangle?
The area of a rectangle is the region it covers in a two-dimensional plane. The area formula is A = Length x Breadth. For example, a rectangle with length 10 cm and breadth 2 cm has an area of 20 square centimetres. The formula applies to all rectangles regardless of size.
Area = Length x Breadth
For example, if the length of a rectangle is A and the breadth is B, then the area is A x B square units.
How Are Length, Breadth and Diagonal of a Rectangle Related?
In a rectangle, the length, breadth, and diagonal form a right-angled triangle. Using the Pythagorean theorem: Diagonal2 = Length2 + Breadth2. This means that if any two of the three values are known, the third can be calculated. This relationship is fundamental to solving practical rectangle problems.
Let us look at some practical examples to understand the area of a rectangle and its properties.
Example 1: The length and breadth of a rectangle are 10 cm and 2 cm respectively. Area = 10 x 2 = 20 cm2.
Example 2: The length of a rectangle is 3 cm and the diagonal is 5 cm. Using Pythagoras: Breadth = square root of (52 - 32) = square root of (25 - 9) = square root of 16 = 4 cm. Area = 3 x 4 = 12 cm2.