Rectangle | Definition, Properties, Formulas
Last Updated :
15 Dec, 2024
A rectangle is a quadrilateral with four sides and following properties.
- All four angles are right angles (90°).
- The opposite sides of a rectangle are equal in length and parallel to each other.
- A rectangle is a two-dimensional flat shape.
Illustration of a Rectangle Here, sides AB and CD are equal and parallel and are called the lengths of the rectangle. Also, the sides BC and AD are called the breadth of the rectangle and they are also parallel and equal. AC is the diagonal of the rectangle.
Read More: Quadrilateral
Length and Breadth of Rectangle
The two sides of the rectangle are classified as the Length and two sides are classified as Breadth.
The longer side of the rectangle is called the length of the rectangle and the shorter side of the rectangle is called the breadth of the rectangle.
From the above image, we have
- Length of the Rectangle ABCD = AB and CD
- Breadth of the Rectangle ABCD = BC and DA
The length and the breadth of the rectangle are used to find the Area, Perimeter, and Length of the Diagonal of the Rectangle.
Diagonal of Rectangle
Diagonal of the rectangle is the line connecting the two points of the rectangle by skipping one point.
In Figure 1 the rectangle ABCD has the diagonal AC and its length can be calculated using the Pythagoras theorem. The diagonal of the rectangle are equal and they bisect each other.
If the length and the breadth of the rectangle are l and b respectively then the diagonal of the rectangle is
d = √( l2 + b2)
Rectangle Properties
Some of the important properties of a rectangle are:
- Rectangle is a Polygon with four sides.
- The angle formed by adjacent sides is 90°.
- Rectangle is a Quadrilateral.
- Opposite sides of a Rectangle are Equal and Parallel.
- Rectangle is considered to be a Parallelogram.
- Interior Angles of a Rectangle are equal and the value of each angle is 90°.
- Sum of all Interior Angles is 360°.
- Diagonals of a Rectangle are equal.
- Diagonals of a Rectangle bisect each other.
- Length of the diagonal is found by Pythagoras theorem. If the length and the breadth of the rectangle are l and b respectively then the diagonal of the rectangle is d = √( l2 + b2).
A rectangle is a closed quadrilateral in which the sides are equal and parallel, there are various formulas that are used to find various parameters of the rectangle.
Some of the important formulas of the rectangle are,
- Perimeter of Rectangle
- Area of Rectangle
| Property | Formula |
|---|
| Area | Length × Breadth |
| Perimeter | 2 × (Length + Breadth) |
Let's discuss them in some detail.
Rectangle Perimeter
Perimeter of the rectangle is defined as the total length of all the sides of the rectangle. It is also called the circumference of the rectangle.
It is measured in units of length, i.e. m, cm, etc.
In a rectangle, if the length(l) and breadth(b) are given then the perimeter of the rectangle is found using the formula,
Perimeter of Rectangle = 2 (l + b)
their
Where,
- l is the length of the rectangle.
- b is the breadth of the rectangle.
The figure representing the perimeter of the rectangle is,
Perimeter of Rectangle IllustrationIn a rectangle, if the breadth(b) and perimeter(P) are given then its length(l) is found using,
(length) l = (P / 2) - b
In a rectangle, if the breadth(b) and area (A) are given then its length(l) is found using,
(length) l = A/b
In a rectangle, if the length(l) and perimeter(P) are given then its breadth(b) is found using,
(breadth) b = (P / 2) - l
In a rectangle, if the length(l) and area (A) are given then its breadth(b) is found using,
(breadth) b = A/l
Rectangle Area
Area of the rectangle is defined as the total space occupied by the rectangle. It is the space inside the boundary or the perimeter of the rectangle.
The area of the rectangle is dependent on the length and breadth of the rectangle, it is the product of the length and the breadth of the rectangle. The area of a rectangle is measured in square units, i.e. in m2, cm2, etc.
In a rectangle, if the length(l) and breadth(b) are given then the area of the rectangle is found using the formula,
Area of Rectangle = l × b
where,
- l is the length of the rectangle
- b is the breadth of the rectangle
Here is a diagram representing the area of the rectangle is,
Area of Rectangle DiagramTypes of Rectangle
Rectangles are of two types, which are:
Let's learn about them in detail.
Square
A square is a special case of a rectangle in which all the sides are also equal. It is a quadrilateral in which opposite sides are parallel and equal, so it can be considered a rectangle.
The diagram of a square is shown below,
Diagram of a SquareLearn More: Square
Golden Rectangle
A rectangle in which the 'length to the width' ratio is similar to the golden ratio, i.e. equal to the ratio of 1: (1+⎷5)/2 is called the golden rectangle.
They are in the ratio of 1: 1.618 thus, if its width is 1 m then its length is 1.168 m.
The diagram of a Golden Rectangle is shown below,
Must Read
Examples on Rectangle
Here are some solved examples on the basic concepts of rectangle for your help.
Example 1: Find the area of the rectangular photo frame whose sides are 8 cm and 6 cm.
Solution:
Given,
- Length of Photo Frame = 8 cm
- Breadth of Photo Frame = 6 cm
Area of Photo Frame = Length × Breadth
= 6 × 8 = 48 cm2
Thus, the area of the photo frame is 48 cm2.
Example 2: Find the perimeter of the rectangular field whose sides are 9 m and 13 m.
Solution:
Given,
- Length of Rectangular Field = 9 m
- Breadth of Rectangular Field = 13 m
Perimeter of Rectangular Field = 2 × (l + b)
= 2 × (9 + 13) = 2 × (22)
= 44 m
Thus, the perimeter of the rectangular field is 44 m.
Example 3: Find the area and the perimeter of the room with a length of 12 feet and a breadth of 8 feet.
Solution:
Given,
- Length of room (l) = 12 feet
- Breadth of room (b) = 8 feet
Area of Room = Length × Breadth
= 12 × 8 = 96 feet2
Perimeter of Room = 2 × (l + b)
= 2 × (12 + 8) = 2 × 20
= 40 feet
Thus, the area of the room is 96 feet2, and the perimeter is 40 feet.
Example 4: Find the length of the diagonal of the rectangle whose sides are 6 cm and 8 cm.
Solution:
Given,
- Length of rectangle (l) = 6 cm
- Breadth of rectangle (b) = 8 cm
Length of diagonal (d) = √( l2 + b2)
d = √( 82 + 62)
d = √ (64 + 36) = √(100)
d = 10 cm
Thus, the length of the rectangle is 10 cm
Similar Reads
Geometry Geometry is a branch of mathematics that studies the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. From basic lines and angles to complex structures, it helps us understand the world around us.Geometry for Students and BeginnersThis section covers key br
2 min read
Lines and Angles
Lines in Geometry- Definition, Types and ExamplesA line in geometry is a straight path that goes on forever in both directions. It has no thickness and is usually drawn between two points, but it keeps going without stopping. Lines are important for making shapes, measuring distances, and understanding angles. For example, the edge of a ruler can
9 min read
Difference between a Line and a RayThe word 'geometry' is the English equivalent of the Greek word 'geometron'. 'Geo' means Earth and 'Metron' means Measure. Even today geometric ideas are reflected in many forms of art, measurement, textile, designing, engineering, etc. All objects have different sizes. For example, the geometry of
3 min read
Angles | Definition, Types and ExamplesIn geometry, an angle is a figure that is formed by two intersecting rays or line segments that share a common endpoint. The word “angle†is derived from the Latin word “angulusâ€, which means “cornerâ€. The two lines joined together are called the arms of the angle and the measure of the opening betw
13 min read
Types of AnglesTypes of Angles: An angle is a geometric figure formed by two rays meeting at a common endpoint. It is measured in degrees or radians. It deals with the relationship of points, lines, angles, and shapes in space. Understanding different types of angles is crucial for solving theoretical problems in
10 min read
2D Geometry
Polygons
Polygon Formula - Definition, Symbol, ExamplesPolygons are closed two-dimensional shapes made with three or more lines, where each line intersects at vertices. Polygons can have various numbers of sides, such as three (triangles), four (quadrilaterals), and more. In this article, we will learn about the polygon definition, the characteristics o
7 min read
Types of PolygonsTypes of Polygons classify all polygons based on various parameters. As we know, a polygon is a closed figure consisting only of straight lines on its edges. In other words, polygons are closed figures made up of more than 2 line segments on a 2-dimensional plane. The word Polygon is made up of two
9 min read
Exterior Angles of a PolygonPolygon is a closed, connected shape made of straight lines. It may be a flat or a plane figure spanned across two-dimensions. A polygon is an enclosed figure that can have more than 3 sides. The lines forming the polygon are known as the edges or sides and the points where they meet are known as ve
6 min read
Triangles
Triangles in GeometryA triangle is a polygon with three sides (edges), three vertices (corners), and three angles. It is the simplest polygon in geometry, and the sum of its interior angles is always 180°. A triangle is formed by three line segments (edges) that intersect at three vertices, creating a two-dimensional re
13 min read
Types of TrianglesA triangle is a polygon with three sides and three angles. It is one of the simplest and most fundamental shapes in geometry. A triangle has these key Properties:Sides: A triangle has three sides, which can have different lengths.Angles: A triangle has three interior angles, and the sum of these ang
5 min read
Angle Sum Property of a TriangleAngle Sum Property of a Triangle is the special property of a triangle that is used to find the value of an unknown angle in the triangle. It is the most widely used property of a triangle and according to this property, "Sum of All the Angles of a Triangle is equal to 180º." Angle Sum Property of a
8 min read
Area of a Triangle | Formula and ExamplesThe area of the triangle is a basic geometric concept that calculates the measure of the space enclosed by the three sides of the triangle. The formulas to find the area of a triangle include the base-height formula, Heron's formula, and trigonometric methods.The area of triangle is generally calcul
6 min read
Theorem - Angle opposite to equal sides of an isosceles triangle are equal | Class 9 MathsIn geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Exampl
4 min read
GRE Geometry | TrianglesTriangle is one of the basic 2-D shapes in geometry. It consists of three angles, three side and three vertices, e.g., On the basis of measurement of sides and angles there different types of triangle listed below: 1. Equilateral triangle: All edges are equal, all angles are also equal to 60°, e.g.,
3 min read
Triangle Inequality Theorem, Proof & ApplicationsTriangle Inequality Theorem is the relation between the sides and angles of triangles which helps us understand the properties and solutions related to triangles. Triangles are the most fundamental geometric shape as we can't make any closed shape with two or one side. Triangles consist of three sid
8 min read
Congruence of Triangles |SSS, SAS, ASA, and RHS RulesCongruence of triangles is a concept in geometry which is used to compare different shapes. It is the condition between two triangles in which all three corresponding sides and corresponding angles are equal. Two triangles are said to be congruent if and only if they can be overlapped with each othe
9 min read
Mid Point TheoremThe Midpoint Theorem is a fundamental concept in geometry that simplifies solving problems involving triangles. It establishes a relationship between the midpoints of two sides of a triangle and the third side. This theorem is especially useful in coordinate geometry and in proving other mathematica
6 min read
X and Y Intercept FormulaX and Y Intercept Formula as the name suggests, is the formula to calculate the intercept of a given straight line. An intercept is defined as the point at which the line or curve intersects the graph's axis. The intercept of a line is the point at which it intersects the x-axis or the y-axis. When
9 min read
Basic Proportionality Theorem (BPT) Class 10 | Proof and ExamplesBasic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT. BPT states that if a line is parallel to a side of a triangl
8 min read
Criteria for Similarity of TrianglesThings are often referred similar when the physical structure or patterns they show have similar properties, Sometimes two objects may vary in size but because of their physical similarities, they are called similar objects. For example, a bigger Square will always be similar to a smaller square. In
9 min read
Pythagoras Theorem | Formula, Proof and ExamplesPythagoras Theorem explains the relationship between the three sides of a right-angled triangle and helps us find the length of a missing side if the other two sides are known. It is also known as the Pythagorean theorem. It states that in a right-angled triangle, the square of the hypotenuse is equ
9 min read
Quadrilateral
Types of Quadrilaterals and Their PropertiesA quadrilateral is a polygon with four sides, four vertices, and four angles. There are several types of quadrilaterals, each with its own properties and characteristics. All different types of quadrilaterals have very distinct shapes and properties which help us identify these quadrilaterals. Types
12 min read
Angle Sum Property of a QuadrilateralAngle Sum Property of a Quadrilateral: Quadrilaterals are encountered everywhere in life, every square rectangle, any shape with four sides is a quadrilateral. We know, three non-collinear points make a triangle. Similarly, four non-collinear points take up a shape that is called a quadrilateral. It
9 min read
Parallelogram | Properties, Formulas, Types, and TheoremA parallelogram is a two-dimensional geometrical shape whose opposite sides are equal in length and are parallel. The opposite angles of a parallelogram are equal in measure and the Sum of adjacent angles of a parallelogram is equal to 180 degrees.A parallelogram is a four-sided polygon (quadrilater
10 min read
Rhombus: Definition, Properties, Formula and ExamplesA rhombus is a type of quadrilateral with the following additional properties. All four sides are of equal length and opposite sides parallel. The opposite angles are equal, and the diagonals bisect each other at right angles. A rhombus is a special case of a parallelogram, and if all its angles are
6 min read
Square in Maths - Area, Perimeter, Examples & ApplicationsA square is a type of quadrilateral where all four sides are of equal length and each interior angle measures 90°. It has two pairs of parallel sides, with opposite sides being parallel. The diagonals of a square are equal in length and bisect each other at right angles.Squares are used in various f
5 min read
Rectangle | Definition, Properties, FormulasA rectangle is a quadrilateral with four sides and following properties. All four angles are right angles (90°). The opposite sides of a rectangle are equal in length and parallel to each other.A rectangle is a two-dimensional flat shape. Illustration of a Rectangle Here, sides AB and CD are equal a
7 min read
Trapezium: Types | Formulas |Properties & ExamplesA Trapezium or Trapezoid is a quadrilateral (shape with 4 sides) with exactly one pair of opposite sides parallel to each other. The term "trapezium" comes from the Greek word "trapeze," meaning "table." It is a two-dimensional shape with four sides and four vertices.In the figure below, a and b are
8 min read
Kite - QuadrilateralsA Kite is a special type of quadrilateral that is easily recognizable by its unique shape, resembling the traditional toy flown on a string. In geometry, a kite has two pairs of adjacent sides that are of equal length. This distinctive feature sets it apart from other quadrilaterals like squares, re
8 min read
Area of Parallelogram | Definition, Formulas & ExamplesA parallelogram is a four-sided polygon (quadrilateral) where opposite sides are parallel and equal in length. In a parallelogram, the opposite angles are also equal, and the diagonals bisect each other (they cut each other into two equal parts).The area of a Parallelogram is the space or the region
8 min read
Euclid’s Geometry
Circle
Circles in MathsA circle is a two-dimensional shape where all points on the circumference are the same distance from the center.A circle consists of all points in a plane that are equidistant (at the same distance) from a fixed point called the centre. The distance from the centre to any point on the circle is call
10 min read
Circumference of Circle - Definition, Perimeter Formula, and ExamplesThe circumference of a circle is the distance around its boundary, much like the perimeter of any other shape. It is a key concept in geometry, particularly when dealing with circles in real-world applications such as measuring the distance traveled by wheels or calculating the boundary of round obj
7 min read
Area of a Circle: Formula, Derivation, ExamplesThe area of a Circle is the measure of the two-dimensional space enclosed within its boundaries. It is mostly calculated by the size of the circle's radius which is the distance from the center of the circle to any point on its edge. The area of a circle is proportional to the radius of the circle.
10 min read
Area of a Circular SectorA circular sector or circle sector is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. Let's look at this figure and try to figure out the sector: source: Wikipedia ( https://goo.gl/mWijn2 ) In this fig
4 min read
Segment of a CircleSegment of a Circle is one of the important parts of the circle other than the sector. As we know, the circle is a 2-D shape in which points are equidistant from the point and the line connecting the two points lying on the circumference of the circle is called the chord of the circle. The area form
7 min read
Circle TheoremsCircle is a collection of points that are at a fixed distance from a particular point. The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle. We come across many objects in real life which are round in shape. For example, wheels of vehicles, ban
5 min read
Tangent to a CircleTangent in Circles are the line segments that touch the given curve only at one particular point. Tangent is a Greek word meaning "To Touch". For a circle, we can say that the line which touches the circle from the outside at one single point on the circumference is called the tangent of the circle.
10 min read
Theorem - The tangent at any point of a circle is perpendicular to the radius through the point of contactA tangent is a straight line drawn from an external point that touches a circle at exactly one point on the circumference of the circle. There can be an infinite number of tangents to a circle. These tangents follow certain properties that can be used as identities to perform mathematical computatio
3 min read
Number of Tangents from a Point on a CircleA circle is a collection of all the points in a plane that are at a constant distance from a particular point. This distance is called the radius of the circle and the fixed point is called the centre. Â A straight line and a circle can co-exist in three ways, one can be a straight line with no inter
11 min read
Theorem - The lengths of tangents drawn from an external point to a circle are equal - Circles | Class 10 MathsTangent is a straight line drawn from an external point that touches a circle at exactly one point on the circumference of the circle. There can be an infinite number of tangents of a circle. These tangents follow certain properties that can be used as identities to perform mathematical computations
5 min read
Equation of a CircleA circle is a geometric shape described as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius. Some key components of the circle are:Center: The fixed point in the middle of the circ
14 min read
What is Cyclic QuadrilateralCyclic Quadrilateral is a special type of quadrilateral in which all the vertices of the quadrilateral lie on the circumference of a circle. In other words, if you draw a quadrilateral and then find a circle that passes through all four vertices of that quadrilateral, then that quadrilateral is call
9 min read
The sum of opposite angles of a cyclic quadrilateral is 180° | Class 9 Maths TheoremIn Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the cir
6 min read
3D Geometry
Visualizing Solid ShapesVisualizing Solid Shapes: Any plane or any shape has two measurements length and width, which is why it is called a two-dimensional(2D) object. Circles, squares, triangles, rectangles, trapeziums, etc. are 2-D shapes. If an object has length, width, and breadth then it is a three-dimensional object(
8 min read
Polyhedron | Meaning, Shapes, Formula, and ExamplesA polyhedron is a 3D solid made up of flat polygonal faces, with edges meeting at vertices. Each face is a polygon, and the edges connect the faces at their vertices. Examples include cubes, prisms, and pyramids. Shapes like cones and spheres are not polyhedrons because they lack polygonal faces.Pol
6 min read
Difference between 2D and 3D Shapes2D shapes are flat like pictures on paper, with just length and breadth but not depth. On the other hand, 3D shapes are like real objects you can touch, with length, breadth, and depth. They take up space, like a toy that you can hold. Examples of 2D shapes include squares and circles. Cubes, sphere
3 min read
Lines
Equation of a Straight Line | Forms, Examples and Practice QuestionsThe equation of a line describes the relationship between the x-coordinates and y-coordinates of all points that lie on the line. It provides a way to mathematically represent that straight path.In general, the equation of a straight line can be written in several forms, depending on the information
10 min read
Slope of a LineSlope of a Line is the measure of the steepness of a line, a surface, or a curve, whichever is the point of consideration. The slope of a Line is a fundamental concept in the stream of calculus or coordinate geometry, or we can say the slope of a line is fundamental to the complete mathematics subje
12 min read
Angle between a Pair of LinesGiven two integers M1 and M2 representing the slope of two lines intersecting at a point, the task is to find the angle between these two lines. Examples: Input: M1 = 1.75, M2 = 0.27Output: 45.1455 degrees Input: M1 = 0.5, M2 = 1.75Output: 33.6901 degrees Approach: If ? is the angle between the two
4 min read
Slope Intercept FormThe slope-intercept formula is one of the formulas used to find the equation of a line. The slope-intercept formula of a line with slope m and y-intercept b is, y = mx + b. Here (x, y) is any point on the line. It represents a straight line that cuts both axes. Slope intercept form of the equation i
9 min read
Point Slope Form Formula of a LineIn geometry, there are several forms to represent the equation of a straight line on the two-dimensional coordinate plane. There can be infinite lines with a given slope, but when we specify that the line passes through a given point then we get a unique straight line. Different forms of equations o
6 min read
Writing Slope-Intercept EquationsStraight-line equations, also known as "linear" equations, have simple variable expressions with no exponents and graph as straight lines. A straight-line equation is one that has only two variables: x and y, rather than variables like y2 or √x. Because it contains information about these two proper
10 min read
Slope of perpendicular to lineYou are given the slope of one line (m1) and you have to find the slope of another line which is perpendicular to the given line. Examples: Input : 5 Output : Slope of perpendicular line is : -0.20 Input : 4 Output : Slope of perpendicular line is : -0.25 Suppose we are given two perpendicular line
3 min read
What is the Point of Intersection of Two Lines Formula?If we consider two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, the point of intersection of these two lines is given by the formula:(x, y) = \left( \frac{b_1 c_2 \ - \ b_2 c_1}{a_1 b_2 \ - \ a_2 b_1}, \frac{c_1 a_2 \ - \ c_2 a_1}{a_1 b_2 \ - \ a_2 b_1} \right),The given illustration shows the i
5 min read
Slope of the line parallel to the line with the given slopeGiven an integer m which is the slope of a line, the task is to find the slope of the line which is parallel to the given line. Examples: Input: m = 2 Output: 2 Input: m = -3 Output: -3 Approach: Let P and Q be two parallel lines with equations y = m1x + b1, and y = m2x + b2 respectively. Here m1 an
3 min read
Minimum distance from a point to the line segment using VectorsGiven the coordinates of two endpoints A(x1, y1), B(x2, y2) of the line segment and coordinates of a point E(x, y); the task is to find the minimum distance from the point to line segment formed with the given coordinates.Note that both the ends of a line can go to infinity i.e. a line has no ending
10 min read
Distance between two parallel linesGiven are two parallel straight lines with slope m, and different y-intercepts b1 & b2.The task is to find the distance between these two parallel lines.Examples: Input: m = 2, b1 = 4, b2 = 3 Output: 0.333333 Input: m = -4, b1 = 11, b2 = 23 Output: 0.8 Approach: Let PQ and RS be the parallel lin
4 min read
Equation of a straight line passing through a point and making a given angle with a given lineGiven four integers a, b, c representing coefficients of a straight line with equation (ax + by + c = 0), the task is to find the equations of the two straight lines passing through a given point (x1, y1) and making an angle ? with the given straight line. Examples: Input: a = 2, b = 3, c = -7, x1 =
15+ min read