Area of Trapezium
A trapezium (also called a trapezoid in some countries) is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases, and the non-parallel sides are called legs.
The formula for the area of a trapezium involves the two parallel sides and the perpendicular distance between them (the height). This formula is an important part of the NCERT Class 8 Mensuration chapter.
The area of a trapezium can be derived by splitting it into simpler shapes (triangles and rectangles) or by combining two congruent trapeziums to form a parallelogram. Both methods give the same result.
What is Area of Trapezium?
Definition: A trapezium is a quadrilateral in which one pair of opposite sides is parallel.
Key Terms:
- Parallel sides (a and b): The two sides that are parallel to each other. They are also called the bases of the trapezium.
- Height (h): The perpendicular distance between the two parallel sides. It is NOT the length of the non-parallel side.
- Legs: The two non-parallel sides of the trapezium.
- Isosceles Trapezium: A trapezium where the two non-parallel sides (legs) are equal in length.
- Right Trapezium: A trapezium with two adjacent right angles.
Area of Trapezium Formula
Area of Trapezium:
Area = ½ × (a + b) × h
Where:
- a = length of one parallel side
- b = length of the other parallel side
- h = perpendicular distance (height) between the parallel sides
Alternative ways to write the formula:
- Area = (Sum of parallel sides × Height) / 2
- Area = (a + b) × h / 2
- Area = ½ × (sum of bases) × height
Units: If sides are in cm and height in cm, area is in cm². If sides are in m and height in m, area is in m².
Derivation and Proof
Derivation of the Area Formula:
Method 1: Using two congruent trapeziums
- Take a trapezium ABCD with parallel sides AB = a (top) and DC = b (bottom), and height = h.
- Make a copy of this trapezium and rotate it 180°.
- Place the rotated copy next to the original so that they fit together to form a parallelogram.
- The base of this parallelogram = a + b.
- The height of the parallelogram = h.
- Area of parallelogram = base × height = (a + b) × h.
- Since the parallelogram is made of two identical trapeziums:
- Area of one trapezium = ½ × (a + b) × h.
Method 2: Splitting into a rectangle and triangles
- Draw a trapezium with parallel sides a (shorter, top) and b (longer, bottom), height h.
- Drop perpendiculars from the ends of the shorter side to the longer side.
- This divides the trapezium into a rectangle of dimensions a × h and two right triangles.
- The sum of the bases of the two triangles = b - a.
- Area of rectangle = a × h.
- Combined area of two triangles = ½ × (b - a) × h.
- Total area = ah + ½(b - a)h = ah + ½h(b - a) = ½h(2a + b - a) = ½h(a + b).
- Area = ½ × (a + b) × h.
Method 3: Using diagonal to split into two triangles
- Draw diagonal AC in trapezium ABCD.
- This creates two triangles: ΔABC and ΔACD.
- Both triangles have the same height h (perpendicular distance between parallel sides).
- Area of ΔABC = ½ × a × h (base = a).
- Area of ΔACD = ½ × b × h (base = b).
- Total area = ½ × a × h + ½ × b × h = ½ × (a + b) × h.
Types and Properties
Problems on area of trapezium can be classified into these types:
1. Finding the area (given parallel sides and height):
- Directly apply Area = ½ × (a + b) × h.
2. Finding the height (given area and parallel sides):
- Rearrange: h = 2 × Area / (a + b).
3. Finding a parallel side (given area, height, and one parallel side):
- Rearrange: a = (2 × Area / h) - b.
4. Trapezium in real-life shapes:
- Cross-section of channels, table tops, land plots, window panes.
5. Combined shapes involving trapezium:
- Finding area of shapes that include trapezium as a part.
6. Unit conversion problems:
- Sides in different units — convert before applying formula.
Solved Examples
Example 1: Example 1: Basic area calculation
Problem: Find the area of a trapezium with parallel sides 12 cm and 8 cm, and height 5 cm.
Solution:
Given:
- a = 12 cm, b = 8 cm, h = 5 cm
Using the formula:
- Area = ½ × (a + b) × h
- = ½ × (12 + 8) × 5
- = ½ × 20 × 5
- = ½ × 100
- = 50 cm²
Answer: The area of the trapezium is 50 cm².
Example 2: Example 2: Finding the height
Problem: The area of a trapezium is 126 cm². The parallel sides are 10 cm and 18 cm. Find the height.
Solution:
Given:
- Area = 126 cm², a = 10 cm, b = 18 cm
Using the formula:
- Area = ½ × (a + b) × h
- 126 = ½ × (10 + 18) × h
- 126 = ½ × 28 × h
- 126 = 14h
- h = 126 / 14
- h = 9 cm
Answer: The height is 9 cm.
Example 3: Example 3: Finding a parallel side
Problem: The area of a trapezium is 200 cm². One parallel side is 16 cm and the height is 10 cm. Find the other parallel side.
Solution:
Given:
- Area = 200 cm², a = 16 cm, h = 10 cm, b = ?
Using the formula:
- 200 = ½ × (16 + b) × 10
- 200 = 5 × (16 + b)
- 16 + b = 200 / 5 = 40
- b = 40 - 16
- b = 24 cm
Answer: The other parallel side is 24 cm.
Example 4: Example 4: Area of a trapezium-shaped field
Problem: A field is in the shape of a trapezium. The parallel sides are 40 m and 60 m, and the perpendicular distance between them is 25 m. Find the area of the field.
Solution:
Given:
- a = 40 m, b = 60 m, h = 25 m
Using the formula:
- Area = ½ × (40 + 60) × 25
- = ½ × 100 × 25
- = 1250 m²
Answer: The area of the field is 1250 m².
Example 5: Example 5: Area with decimal measurements
Problem: Find the area of a trapezium with parallel sides 7.5 cm and 12.5 cm, and height 6.4 cm.
Solution:
Given:
- a = 7.5 cm, b = 12.5 cm, h = 6.4 cm
Using the formula:
- Area = ½ × (7.5 + 12.5) × 6.4
- = ½ × 20 × 6.4
- = 10 × 6.4
- = 64 cm²
Answer: The area is 64 cm².
Example 6: Example 6: Cross-section of a canal
Problem: The cross-section of a canal is in the shape of a trapezium. The top width is 12 m, the bottom width is 8 m, and the depth is 3 m. Find the area of the cross-section.
Solution:
Given:
- a = 12 m (top), b = 8 m (bottom), h = 3 m (depth)
Using the formula:
- Area = ½ × (12 + 8) × 3
- = ½ × 20 × 3
- = 30 m²
Answer: The area of the cross-section is 30 m².
Example 7: Example 7: Comparing areas of two trapeziums
Problem: Trapezium A has parallel sides 10 cm and 14 cm with height 6 cm. Trapezium B has parallel sides 8 cm and 16 cm with height 6 cm. Which has greater area?
Solution:
Trapezium A:
- Area = ½ × (10 + 14) × 6 = ½ × 24 × 6 = 72 cm²
Trapezium B:
- Area = ½ × (8 + 16) × 6 = ½ × 24 × 6 = 72 cm²
Answer: Both trapeziums have equal area (72 cm²). This is because the sum of parallel sides (24 cm) and the height (6 cm) are the same in both.
Example 8: Example 8: Area of a table top
Problem: A table top is in the shape of a trapezium. The parallel edges are 1.2 m and 0.8 m, and the distance between them is 0.6 m. Find the area in cm².
Solution:
Step 1: Convert to cm
- a = 1.2 m = 120 cm
- b = 0.8 m = 80 cm
- h = 0.6 m = 60 cm
Step 2: Find area
- Area = ½ × (120 + 80) × 60
- = ½ × 200 × 60
- = 6000 cm²
Answer: The area of the table top is 6000 cm².
Example 9: Example 9: Ratio of parallel sides
Problem: The parallel sides of a trapezium are in the ratio 3:5. If the height is 8 cm and the area is 96 cm², find the parallel sides.
Solution:
Given:
- Parallel sides = 3x and 5x, h = 8 cm, Area = 96 cm²
Using the formula:
- 96 = ½ × (3x + 5x) × 8
- 96 = ½ × 8x × 8
- 96 = 32x
- x = 3
Parallel sides:
- 3x = 3 × 3 = 9 cm
- 5x = 5 × 3 = 15 cm
Answer: The parallel sides are 9 cm and 15 cm.
Example 10: Example 10: Cost of laying grass on trapezium-shaped land
Problem: A garden is in the shape of a trapezium with parallel sides 30 m and 20 m, and height 15 m. The cost of laying grass is Rs 50 per m². Find the total cost.
Solution:
Step 1: Find the area
- Area = ½ × (30 + 20) × 15
- = ½ × 50 × 15
- = 375 m²
Step 2: Find the cost
- Cost = Area × Rate
- = 375 × 50
- = Rs 18,750
Answer: The total cost of laying grass is Rs 18,750.
Real-World Applications
Land Measurement: Many plots of land, agricultural fields, and gardens are trapezium-shaped. Surveyors use the trapezium area formula to calculate land area.
Civil Engineering: Cross-sections of canals, dams, and embankments are often trapezoidal. Engineers calculate the cross-sectional area for water flow and material estimates.
Architecture: Trapezium-shaped windows, doors, and roof sections require area calculation for glass, paint, or material estimation.
Everyday Objects: Table tops, bucket shapes (cross-sections), handbags, and ramps often have trapezoidal cross-sections.
Road Design: Road cross-sections are trapezoidal — wider at the top and narrower at the drainage base. Area calculation helps in estimating material for road construction.
Key Points to Remember
- A trapezium has exactly one pair of parallel sides.
- The area formula is: Area = ½ × (a + b) × h, where a and b are parallel sides and h is the height.
- The height is the perpendicular distance between the parallel sides, NOT the slant side.
- The formula can be derived by combining two trapeziums to form a parallelogram.
- If the sum of parallel sides is the same and the height is the same, the areas are equal (regardless of individual side lengths).
- To find height: h = 2 × Area / (a + b).
- To find a parallel side: a = (2 × Area / h) - b.
- Units of area are always square units (cm², m²).
- The area of a trapezium is always between the areas of two parallelograms: one with base a and one with base b, both with height h.
- When both parallel sides are equal (a = b), the trapezium becomes a parallelogram, and the formula gives Area = a × h.
Practice Problems
- Find the area of a trapezium with parallel sides 15 cm and 9 cm, and height 7 cm.
- The area of a trapezium is 180 cm². If the parallel sides are 14 cm and 22 cm, find the height.
- The parallel sides of a trapezium are in the ratio 4:7 and the height is 12 cm. If the area is 264 cm², find the parallel sides.
- A trapezium-shaped field has parallel sides 50 m and 70 m with a perpendicular distance of 40 m. Find the cost of ploughing at Rs 30 per m².
- One parallel side of a trapezium is twice the other. If the height is 10 cm and the area is 150 cm², find both parallel sides.
- The cross-section of a swimming pool is a trapezium with parallel sides 2 m and 4 m and depth 1.5 m. Find the area of the cross-section.
- Find the height of a trapezium whose area is 420 m² and parallel sides are 25 m and 35 m.
- A trapezoidal garden has area 540 m². One parallel side is 30 m and the height is 18 m. Find the other parallel side.
Frequently Asked Questions
Q1. What is the formula for area of a trapezium?
Area of trapezium = ½ × (sum of parallel sides) × height = ½ × (a + b) × h, where a and b are the two parallel sides and h is the perpendicular distance between them.
Q2. What is the height of a trapezium?
The height (h) of a trapezium is the perpendicular distance between the two parallel sides. It is NOT the length of the non-parallel side (leg). The height is always measured at right angles to the parallel sides.
Q3. How is the area formula derived?
Take two identical trapeziums and place them together (one rotated 180°) to form a parallelogram with base (a + b) and height h. Area of parallelogram = (a + b) × h. Since it contains two trapeziums, area of one trapezium = ½ × (a + b) × h.
Q4. What is the difference between a trapezium and a parallelogram?
A trapezium has exactly ONE pair of parallel sides. A parallelogram has TWO pairs of parallel sides. When both pairs of opposite sides of a trapezium become parallel, it becomes a parallelogram.
Q5. Can the area of a trapezium be zero?
The area is zero only if the height is zero (the parallel sides coincide) or both parallel sides are zero. In a proper trapezium, the area is always positive.
Q6. Is the area formula the same for all types of trapeziums?
Yes. The formula Area = ½ × (a + b) × h works for all trapeziums — right trapezium, isosceles trapezium, or any irregular trapezium. Only the parallel sides and height matter.
Q7. What happens to the formula when a = b?
When a = b, the trapezium becomes a parallelogram (or rectangle). The formula gives Area = ½ × (a + a) × h = ½ × 2a × h = a × h, which is the area of a parallelogram.
Q8. How do you find a missing parallel side?
Rearrange the formula: a = (2 × Area / h) - b. Substitute the known values of area, height, and one parallel side to find the other.
Q9. What is an isosceles trapezium?
An isosceles trapezium is a trapezium where the two non-parallel sides (legs) are equal in length. Its base angles are equal, and the diagonals are equal in length.
Q10. What are the units of area of a trapezium?
The area is always in square units. If the sides and height are in cm, the area is in cm². If in metres, the area is in m². Always ensure all measurements are in the same unit before calculating.
