Properties of Rectangle
A rectangle is one of the most common shapes in everyday life. Doors, books, mobile screens, blackboards, and table tops are all rectangular.
In geometry, a rectangle is a special type of parallelogram in which every angle is a right angle (90°). This means a rectangle inherits all the properties of a parallelogram and has additional properties of its own.
In Class 8 NCERT, rectangles are studied as part of the chapter "Understanding Quadrilaterals". You will learn how a rectangle differs from other parallelograms like rhombus and square, and how its diagonals behave.
Understanding the properties of a rectangle is essential for solving problems on area, perimeter, diagonals, and coordinate geometry in higher classes.
What is Properties of Rectangle?
Definition: A rectangle is a quadrilateral in which all four interior angles are right angles (each equal to 90°).
Equivalently, a rectangle is a parallelogram with one right angle (since opposite angles are equal in a parallelogram, one right angle forces all four angles to be 90°).
Key features:
- It has 4 sides and 4 vertices.
- Opposite sides are equal and parallel.
- All four angles are 90°.
- The diagonals are equal in length and bisect each other.
- A rectangle is always a parallelogram, but a parallelogram is not always a rectangle.
Standard notation: A rectangle ABCD has vertices A, B, C, D. Sides AB and CD are one pair of opposite sides. Sides BC and DA are the other pair. Diagonals are AC and BD.
Properties of Rectangle Formula
Formulas for a Rectangle:
Perimeter = 2(length + breadth) = 2(l + b)
Area = length × breadth = l × b
Diagonal = √(l² + b²)
Where:
- l = length of the rectangle (longer side)
- b = breadth of the rectangle (shorter side)
Note: The diagonal formula comes from the Pythagoras Theorem, since each diagonal divides the rectangle into two right-angled triangles.
Derivation and Proof
Why all angles of a rectangle are 90°:
- By definition, a rectangle has all angles equal to 90°.
- Since the sum of interior angles of a quadrilateral is 360°, and all four angles are equal: each angle = 360° ÷ 4 = 90°.
Why opposite sides are equal:
- A rectangle is a parallelogram (opposite sides are parallel by definition).
- In any parallelogram, opposite sides are equal.
- Therefore, in rectangle ABCD: AB = CD and BC = DA.
Why diagonals are equal:
- In rectangle ABCD, consider triangles ABC and DCB.
- AB = DC (opposite sides of rectangle).
- BC = BC (common side).
- ∠ABC = ∠DCB = 90° (all angles are 90°).
- By SAS congruence: △ABC ≅ △DCB.
- Therefore, AC = DB (corresponding parts of congruent triangles).
- So the diagonals of a rectangle are equal.
Why diagonals bisect each other:
- A rectangle is a parallelogram.
- In every parallelogram, the diagonals bisect each other.
- Therefore, in a rectangle, the diagonals bisect each other.
- If the diagonals intersect at point O, then AO = OC and BO = OD.
Diagonal length using Pythagoras Theorem:
- In rectangle ABCD with length l and breadth b, diagonal AC forms a right triangle with sides AB = l and BC = b.
- By Pythagoras Theorem: AC² = AB² + BC² = l² + b²
- Therefore: AC = √(l² + b²)
Types and Properties
A rectangle is a specific type of quadrilateral. Here is how it relates to other quadrilaterals:
1. Rectangle as a Parallelogram:
- Every rectangle is a parallelogram (opposite sides parallel and equal).
- A rectangle is a parallelogram with the extra condition that all angles are 90°.
2. Rectangle vs Square:
- A square is a special rectangle where all four sides are equal.
- Every square is a rectangle, but not every rectangle is a square.
- In a square, diagonals bisect each other at 90°. In a rectangle (that is not a square), diagonals bisect each other but NOT at 90°.
3. Rectangle vs Rhombus:
- A rhombus has all sides equal but angles are not necessarily 90°.
- A rectangle has all angles 90° but adjacent sides are not necessarily equal.
- A shape that is both a rectangle and a rhombus is a square.
4. Rectangle vs Parallelogram:
- Both have opposite sides equal and parallel.
- Rectangle has all angles 90°; a general parallelogram does not.
- Rectangle has equal diagonals; a general parallelogram does not.
Summary Table:
- Parallelogram: Opposite sides equal and parallel, opposite angles equal, diagonals bisect each other.
- Rectangle: All parallelogram properties + all angles 90° + diagonals equal.
- Square: All rectangle properties + all sides equal + diagonals perpendicular.
Solved Examples
Example 1: Example 1: Finding the perimeter
Problem: Find the perimeter of a rectangle with length 15 cm and breadth 8 cm.
Solution:
Given:
- Length (l) = 15 cm
- Breadth (b) = 8 cm
Using the formula:
- Perimeter = 2(l + b)
- = 2(15 + 8)
- = 2 × 23
- = 46 cm
Answer: The perimeter is 46 cm.
Example 2: Example 2: Finding the area
Problem: Find the area of a rectangle with length 12 m and breadth 7 m.
Solution:
Given:
- Length (l) = 12 m
- Breadth (b) = 7 m
Using the formula:
- Area = l × b
- = 12 × 7
- = 84 m²
Answer: The area is 84 m².
Example 3: Example 3: Finding the diagonal
Problem: Find the length of the diagonal of a rectangle with length 9 cm and breadth 12 cm.
Solution:
Given:
- Length (l) = 9 cm
- Breadth (b) = 12 cm
Using the formula:
- Diagonal = √(l² + b²)
- = √(9² + 12²)
- = √(81 + 144)
- = √225
- = 15 cm
Answer: The diagonal is 15 cm.
Example 4: Example 4: Finding breadth from area and length
Problem: The area of a rectangle is 180 cm² and its length is 20 cm. Find the breadth.
Solution:
Given:
- Area = 180 cm²
- Length (l) = 20 cm
Using: Area = l × b
- 180 = 20 × b
- b = 180 ÷ 20
- b = 9 cm
Answer: The breadth is 9 cm.
Example 5: Example 5: Finding length from perimeter and breadth
Problem: The perimeter of a rectangle is 56 cm and the breadth is 10 cm. Find the length.
Solution:
Given:
- Perimeter = 56 cm
- Breadth (b) = 10 cm
Using: Perimeter = 2(l + b)
- 56 = 2(l + 10)
- 28 = l + 10
- l = 28 − 10
- l = 18 cm
Answer: The length is 18 cm.
Example 6: Example 6: Verifying diagonal property
Problem: In rectangle PQRS, diagonal PR = 13 cm. Find the length of diagonal QS.
Solution:
Given:
- PR = 13 cm
Property: In a rectangle, both diagonals are equal.
- Therefore, QS = PR = 13 cm
Answer: QS = 13 cm.
Example 7: Example 7: Finding diagonal halves
Problem: The diagonals of a rectangle are 10 cm each. They intersect at point O. Find OA and OB.
Solution:
Given:
- Diagonal AC = BD = 10 cm
Property: Diagonals of a rectangle bisect each other.
- OA = OC = AC ÷ 2 = 10 ÷ 2 = 5 cm
- OB = OD = BD ÷ 2 = 10 ÷ 2 = 5 cm
Answer: OA = OB = OC = OD = 5 cm.
Note: In a rectangle, O is equidistant from all four vertices. This means a circle with centre O and radius = half-diagonal passes through all four vertices.
Example 8: Example 8: Verifying if a quadrilateral is a rectangle
Problem: A parallelogram has one angle equal to 90°. Is it a rectangle?
Solution:
Given: ABCD is a parallelogram with ∠A = 90°.
Step 1: In a parallelogram, opposite angles are equal.
- ∠A = ∠C = 90°
Step 2: Adjacent angles of a parallelogram are supplementary (sum = 180°).
- ∠A + ∠B = 180°
- 90° + ∠B = 180°
- ∠B = 90°
Step 3: ∠B = ∠D = 90° (opposite angles).
All four angles are 90°.
Answer: Yes, the parallelogram is a rectangle.
Example 9: Example 9: Using diagonals to find sides
Problem: The diagonal of a rectangle is 25 cm and one side is 7 cm. Find the other side.
Solution:
Given:
- Diagonal (d) = 25 cm
- One side = 7 cm
Using Pythagoras Theorem:
- d² = l² + b²
- 25² = l² + 7²
- 625 = l² + 49
- l² = 625 − 49 = 576
- l = √576 = 24 cm
Answer: The other side is 24 cm.
Example 10: Example 10: Area and perimeter word problem
Problem: A rectangular garden is 30 m long and 20 m wide. Find the cost of fencing the garden at ₹25 per metre. Also find the area of the garden.
Solution:
Given:
- Length (l) = 30 m, Breadth (b) = 20 m
- Cost of fencing = ₹25 per metre
Step 1: Perimeter
- Perimeter = 2(l + b) = 2(30 + 20) = 2 × 50 = 100 m
Step 2: Cost of fencing
- Cost = Perimeter × Rate = 100 × 25 = ₹2,500
Step 3: Area
- Area = l × b = 30 × 20 = 600 m²
Answer: Cost of fencing = ₹2,500. Area = 600 m².
Real-World Applications
Architecture and Construction: Most rooms, doors, windows, and building foundations are rectangular. Knowing rectangle properties helps in calculating area for flooring, perimeter for fencing, and diagonals for structural support.
Everyday Objects: Books, screens (TV, laptop, mobile), tables, and photograph frames are rectangular. Understanding their proportions uses rectangle properties.
Coordinate Geometry: Rectangles on the coordinate plane are used to find distances, midpoints, and verify perpendicularity. The diagonal property helps confirm whether four points form a rectangle.
Area and Perimeter Problems: Rectangle formulas are used in calculating the cost of tiling, fencing, painting walls, and wrapping gifts.
Sports: Cricket pitches, football fields, basketball courts, and swimming pools are rectangular. Dimensions and diagonal measurements use rectangle properties.
Computer Graphics: Screens are rectangular. Image resolution, pixel dimensions, and aspect ratios all relate to rectangle properties.
Key Points to Remember
- A rectangle is a parallelogram in which all four angles are 90°.
- Opposite sides are equal and parallel: AB = CD and BC = DA.
- All four angles are right angles (90° each).
- Diagonals are equal: AC = BD.
- Diagonals bisect each other: OA = OC and OB = OD (where O is the intersection point).
- Diagonals do NOT bisect each other at right angles (unlike a rhombus or square).
- Perimeter = 2(l + b).
- Area = l × b.
- Diagonal = √(l² + b²) (by Pythagoras Theorem).
- A parallelogram with ONE right angle is a rectangle (all angles become 90°).
- A parallelogram with equal diagonals is a rectangle.
- Every square is a rectangle, but not every rectangle is a square.
Practice Problems
- Find the perimeter and area of a rectangle with length 25 cm and breadth 14 cm.
- The diagonal of a rectangle is 20 cm and one side is 16 cm. Find the other side.
- A rectangular field has a perimeter of 120 m. If the length is twice the breadth, find both dimensions.
- The diagonals of rectangle ABCD intersect at O. If AC = 18 cm, find OA, OB, OC, and OD.
- Prove that the diagonals of a rectangle are equal using congruent triangles.
- A rectangle has length 24 cm and diagonal 26 cm. Find its breadth, perimeter, and area.
- Is a square always a rectangle? Is a rectangle always a square? Justify your answer.
- The length and breadth of a rectangle are in the ratio 5 : 3. If the perimeter is 96 cm, find the dimensions.
Frequently Asked Questions
Q1. What is a rectangle?
A rectangle is a quadrilateral (four-sided polygon) in which all four interior angles are right angles (90° each). Its opposite sides are equal and parallel.
Q2. Is a rectangle a parallelogram?
Yes. A rectangle is a special type of parallelogram where all angles are 90°. It has all the properties of a parallelogram plus the additional property that all angles are right angles and diagonals are equal.
Q3. Are the diagonals of a rectangle equal?
Yes. The diagonals of a rectangle are always equal in length. This can be proved using the SAS congruence rule on the two triangles formed by a diagonal.
Q4. Do the diagonals of a rectangle bisect each other?
Yes. Since a rectangle is a parallelogram, and diagonals of a parallelogram always bisect each other, the diagonals of a rectangle also bisect each other.
Q5. Do the diagonals of a rectangle bisect each other at right angles?
No. The diagonals of a rectangle bisect each other, but NOT at right angles (unless the rectangle is a square). In a rhombus or square, diagonals bisect at 90°.
Q6. What is the difference between a rectangle and a square?
A square has all four sides equal, while a rectangle has only opposite sides equal. Every square is a rectangle, but not every rectangle is a square. A square's diagonals also bisect each other at 90°, while a rectangle's diagonals do not.
Q7. How do you find the diagonal of a rectangle?
Use the Pythagoras Theorem. Diagonal = √(length² + breadth²). The diagonal forms the hypotenuse of a right triangle with the length and breadth as the other two sides.
Q8. How do you prove a parallelogram is a rectangle?
A parallelogram is a rectangle if any ONE of these is true: (1) one angle is 90°, (2) the diagonals are equal in length, (3) it can be inscribed in a circle with the diagonal as diameter.
Q9. What is the angle sum of a rectangle?
The sum of all interior angles of a rectangle is 360°. Since each angle is 90°: 90° + 90° + 90° + 90° = 360°.
Q10. Can a rectangle have unequal adjacent sides?
Yes. In a rectangle, adjacent sides are generally unequal (length ≠ breadth). If all four sides are equal, the rectangle becomes a square.
