Area and Perimeter of Quadrilaterals

To fence a garden, we calculate its perimeter. To cover a floor with tiles, we calculate its area. In this lesson, we will learn quadrilateral formulas through simple examples.

👉 What Is a Quadrilateral?

Quadrilateral

A quadrilateral is a simple plane figure enclosedThe line segments join to form a closed boundary. by 4 straight line segments. Its sides intersect only at their endpoints. In both diagrams above, ABCD is a quadrilateral.

A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals. The diagonals shown are AC and BD.

We will study the area and perimeter of different types of quadrilaterals.

Examples include squares, rectangles, parallelograms, rhombuses and trapeziums.

👉 Perimeter

The perimeter of a closed figure is the total length of its boundary.

Perimeter = AB + BC + CD + DA

Perimeter is measured in units of length, such as cm or m.

👉 Area

Area measures the region enclosed by a plane figure.

The area of quadrilateral ABCD measures the region enclosed by its sides.

Area is measured in square units, such as cm², m².

👉 Square

A square has 4 equal sides and every angle is 90°. Equal sides alone do not make a quadrilateral a square; its angles must also be right angles.

All sides of a square are equal

Each side has length a units

🔸 Perimeter = 4 × side length = 4a units

🔸 Diagonal =√2 × side length = √2a units

🔸 Area = (side length)² = a² square units

📚 Area of a square = × (diagonal)² square units

⭐ Area of a Path around a Square

These formulas apply to a uniform path with square inner and outer boundaries. d is the path width; d does not denote a diagonal here. For an inner path with a square remaining in the middle, .

🔸 For an inner path of width d, area 4d × (a − d) square units.

🔸 For an outer path of width d, area 4d × (a + d) square units.

✍️ Square Practice Problems

🔸 1. A square has side length 9 cm. Find its area and perimeter.

🔸 2. A square has diagonal 2√2 m. Find its area.

🔸 3. A square has area 169 m². Find its diagonal.

🔸 4. A square garden has area 225 m². Find the cost of fencing its entire boundary at ₹3.50 per metre.

🔸 5. The total area of 2 squares is 325 m² and their sides are in the ratio 2:3. Find the sum of their perimeters.

🔸 6. Reducing a square's side by 2 cm decreases its area by 60 cm². Find its original perimeter.

👉 Rectangle

A rectangle has 4 right angles. Its opposite sides are equal and parallel. A square is a special rectangle with all sides equal.

Opposite sides of a rectangle are equal

Length = a units and width = b units

🔸 Perimeter = 2 × (length + width) = 2(a + b) units

🔸 Area = (length × width) = ab square units

🔸 Diagonal units

⭐ Area of a Path around a Rectangle

These formulas apply to a uniform path with rectangular inner and outer boundaries. For an inner path, d > 0 and .

🔸 For an inner path of width d, area 2d × (a + b − 2d) square units.

🔸 For an outer path of width d, area 2d × (a + b + 2d) square units.

✍️ Rectangle Practice Problems

🔸 1. A rectangle is 18 cm long and 11 cm wide. Find its area.

🔸 2. A rectangle has diagonal 13 cm and one side 12 cm. Find its area.

🔸 3. A rectangle's sides are in the ratio 8:5 and its area is 360 m². Find its perimeter.

🔸 4. A square and a rectangle have equal diagonals. The rectangle's length is twice its width. Find the ratio of the square's area to the rectangle's area.

Parallelogram

A parallelogram has 2 pairs of parallel opposite sides. Let adjacent sides be a and b, base b and its corresponding perpendicular height h. The height is the perpendicular distance between the base and the opposite side.

Opposite sides of a parallelogram are equal

Adjacent sides a and b; base b and corresponding perpendicular height h

🔸 Perimeter = 2 × (sum of adjacent side lengths) = 2(a + b) units

🔸 Area = (base × corresponding perpendicular height) = bh square units

Remember: A sloping side is not the perpendicular height. For base b, the area is bh, not ab, unless the included angle is 90°.

Why? Cutting and rearranging suitable pieces of a parallelogram forms a rectangle with the same base and perpendicular height. Rearrangement preserves area, so area = base × height.

Rhombus

A rhombus is a parallelogram with 4 equal sides. Its angles need not be right angles. A square is a special rhombus.

All sides of a rhombus are equal

Each side has length a and corresponding perpendicular height h

🔸 Perimeter = 4 × side length = 4a units

🔸 Area = (side × corresponding perpendicular height) = ah square units

⭐ Rhombus Area Using Diagonals

The diagonals of a rhombus bisect each other at right angles. If their lengths are and , then

Why? The diagonals divide the rhombus into 4 right triangles. Adding their areas gives half the product of the diagonals.

Trapezium

Here, a trapezium means a quadrilateral with exactly 1 pair of parallel opposite sides. Let the parallel sides have lengths a and b, and let the perpendicular distance between them be h.

A trapezium has one pair of parallel opposite sides

Parallel sides a and b; perpendicular distance h between them

🔸 Area = 1/2 (a + b)h square units

If the other sides are c and e, perimeter a + b + c + e units. The parallel sides and height alone do not determine the perimeter.

Why? A diagonal divides the trapezium into 2 triangles with bases a and b and the same height h. Adding their areas gives the formula.

👉 Kite

A convex kite has 2 distinct pairs of equal adjacent sides. Its diagonals are perpendicular.

If the equal pairs have lengths a, a and b, b, perimeter 2(a + b).

If the diagonal lengths are and , then

square units.

Remember: Half the product of the diagonals gives the area when the diagonals are perpendicular. This formula does not apply to every quadrilateral.

👉 Area of a General Convex Quadrilateral

In convex quadrilateral ABCD, let diagonal AC have length k. From opposite vertices B and D, draw perpendiculars to AC or its extension, with lengths and .

The diagonal divides the quadrilateral into 2 triangles. Adding their areas gives this formula. The lengths of 4 sides alone generally do not determine a quadrilateral's area.

👉 Simple Worked Examples

✏️ Example 1: Square

A square has side length 5 cm. Find its area and perimeter.

Solution: Area 5² = 25 cm²; perimeter 4 × 5 = 20 cm.

Answer: Area 25 cm²; perimeter 20 cm.

✏️ Example 2: Inner Path in a Square Garden

A square garden has side length 10 m. A path 1 m wide runs inside its boundary. Find the path's area.

Solution: Side of the remaining inner square 10 − 2 × 1 = 8 m.

Path area 10² − 8² = 100 − 64 = 36 m².

Using the formula also gives 4d(a − d) = 4 × 1 × (10 − 1) = 36 m².

Answer: 36 m².

✏️ Example 3: Rectangle Area, Perimeter and Diagonal

A rectangle is 4 cm long and 3 cm wide. Find its area, perimeter and diagonal.

Solution: Area 4 × 3 = 12 cm².

Perimeter 2(4 + 3) = 14 cm.

Diagonal cm.

Answer: Area 12 cm², perimeter 14 cm, diagonal 5 cm.

✏️ Example 4: Outer Path around a Rectangle

A rectangular garden is 8 m long and 5 m wide. A path 1 m wide runs outside it, with a rectangular outer boundary. Find the path's area.

Solution: Including the path, length 8 + 2 = 10 m and width 5 + 2 = 7 m.

Path area 10 × 7 − 8 × 5 = 70 − 40 = 30 m².

Using the formula also gives 2 × 1 × (8 + 5 + 2 × 1) = 30 m².

Answer: 30 m².

✏️ Example 5: Parallelogram

A parallelogram has base 8 cm, adjacent side 5 cm and perpendicular height 4 cm. Find its area and perimeter.

Solution: Area 8 × 4 = 32 cm².

Perimeter 2(8 + 5) = 26 cm.

Answer: Area 32 cm²; perimeter 26 cm. Use height 4 cm, not the sloping side 5 cm, to find the area.

✏️ Example 6: Rhombus

A rhombus has diagonals 6 cm and 8 cm. Find its area and perimeter.

Solution: Area (1/2) × 6 × 8 = 24 cm².

The diagonals bisect each other at right angles, so the side length is

cm.

Perimeter 4 × 5 = 20 cm.

Answer: Area 24 cm²; perimeter 20 cm.

✏️ Example 7: Trapezium

A trapezium has parallel sides 8 cm and 12 cm and height 5 cm. Find its area.

Solution:

cm².

Answer: 50 cm².

✏️ Example 8: Kite

A convex kite has diagonals 10 cm and 6 cm. Find its area.

Solution: Area (1/2) × 10 × 6 = 30 cm².

Answer: 30 cm².

✅ Answers and Short Solutions to the Earlier Practice Problems

Square Practice Answers

🔸 1. Area 9² = 81 cm²; perimeter 4 × 9 = 36 cm.

🔸 2. Area m².

🔸 3. Side √169 = 13 m; diagonal m.

🔸 4. Side √225 = 15 m; perimeter 60 m; cost 60 × 3.50 = 210 rupees.

🔸 5. Let the sides be 2x and 3x m. Then 4x² + 9x² = 325, so x² = 25 and x = 5. The sides are 10 m and 15 m. Sum of perimeters 4(10 + 15) = 100 m.

🔸 6. Let the original side be a cm. Then a² − (a − 2)² = 60, so 4a − 4 = 60 and a = 16. Perimeter 4 × 16 = 64 cm.

Rectangle Practice Answers

🔸 1. Area 18 × 11 = 198 cm².

🔸 2. Other side cm; area 12 × 5 = 60 cm².

🔸 3. Let the dimensions be 8x and 5x m. Then 40x² = 360, giving x = 3. The dimensions are 24 m and 15 m; perimeter 2(24 + 15) = 78 m.

🔸 4. Let the rectangle's width be x and length 2x. Its diagonal squared is 5x². Square area 5x²/2; rectangle area 2x². Required ratio 5:4.

✍️ Try These Yourself

🔸 1. Find the area of a parallelogram with base 9 cm and height 4 cm.

🔸 2. Find the area of a rhombus with diagonals 10 cm and 12 cm.

🔸 3. Find the area of a trapezium with parallel sides 6 cm and 10 cm and height 3 cm.

🔸 4. A convex quadrilateral has a diagonal of 10 cm. The perpendicular distances of the other 2 vertices from it are 3 cm and 4 cm. Find its area.

Answers: 1. 36 cm²; 2. 60 cm²; 3. 24 cm²; 4. 35 cm².

🧠 Important Points to Remember

• Perimeter and area: Perimeter measures boundary length; area measures the enclosed region.
• Units: Use length units for perimeter and square units for area. Convert all measurements to compatible units before calculating.
• Height: Use the perpendicular height corresponding to the chosen base, not a sloping side.
• Diagonal formula: Half the product of diagonals gives the area of a rhombus or convex kite; it does not work for every quadrilateral.
• Inner path: Subtract 2d from each original dimension because the path occupies both sides.
• Outer path: For square or rectangular outer boundaries, add 2d to each original dimension. The same formula does not apply directly to paths with rounded corners.
• Fencing cost: Multiply total fencing length by the cost per unit length. Subtract any gate opening before calculating.

💡 Quick Revision

Here, h is the corresponding perpendicular height and are the diagonal lengths.