Area of Trapezium with Parallel Sides and Height

Area of Trapezium is calculated using the average of its two parallel sides and the perpendicular distance between them. This page explains the formula, direct calculations, methods for finding missing dimensions, unit conversions and solved numerical examples for quantitative aptitude.

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What Is the Area of a Trapezium?

The area of a trapezium is half the sum of its parallel sides multiplied by the perpendicular height. If the parallel sides are a and b and the height is h, then A = ½(a + b)h.

The two parallel sides are called the bases. The height is the shortest distance measured at a right angle between these bases, not the length of either slanting side. For example, if a = 8 cm, b = 14 cm and h = 5 cm, then A = ½(8 + 14) × 5 = 55 cm².

Area of Trapezium Video Lessons

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Area of a Trapezium

Learn how to calculate the area of a trapezium using its parallel sides and height, with the standard mensuration formula explained clearly.

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Area of Trapezium Formula & Tricks

Important Formulas

Area of a trapezium
A = ½(a + b)h

Here, a and b are the parallel sides and h is the perpendicular height.

Height of a trapezium
h = 2A ÷ (a + b)

Use this form when the area and both parallel sides are known.

Unknown parallel side
b = (2A ÷ h) − a

Use this rearrangement when the area, height and one parallel side are given.

Average-base form
A = average of parallel sides × height = [(a + b) ÷ 2] × h

The area equals the average length of the two parallel sides multiplied by the perpendicular height.

Quick Tricks

Use the average of the parallel sides

First add the two parallel sides and divide their sum by 2. Then multiply by the perpendicular height. This avoids errors with the factor ½.

Example: For parallel sides 12 m and 18 m and height 7 m: average = 15 m, so area = 15 × 7 = 105 m².
Split the trapezium when dimensions are irregular

Drop perpendiculars from the ends of the shorter base to the longer base. The figure can then be treated as a rectangle and one or two right triangles. Add their areas to verify the trapezium formula.

Example: A trapezium with bases 10 cm and 16 cm and height 4 cm has area ½(10 + 16) × 4 = 52 cm².

Area of Trapezium Concepts

Identifying the parallel sides and height

The bases in the formula are the parallel sides, while the height is the perpendicular distance between them.

The height is not generally equal to a sloping side. If a diagram gives an oblique side, use it as the height only when it is perpendicular to the two parallel sides. The area is measured in square units.

Example: If the parallel sides are 9 cm and 15 cm and the perpendicular distance is 6 cm, the slanting side is not needed for area: A = ½(9 + 15) × 6 = 72 cm².

Calculating area from three dimensions

To find the area, add the parallel sides, divide by 2, and multiply by the perpendicular height.

Use consistent units for all three lengths before substituting. If the sides are measured in metres and the height is measured in centimetres, convert one unit so that all lengths match.

Example: For a = 7 m, b = 13 m and h = 4 m, A = ½(7 + 13) × 4 = 40 m².

Finding a missing dimension

Rearrange A = ½(a + b)h before substituting values for a missing height or parallel side.

For a missing height, use h = 2A/(a + b). For a missing parallel side, use b = 2A/h − a. The resulting length must have the same unit as the given sides.

Example: If A = 96 cm², a = 10 cm and b = 14 cm, then h = 2 × 96 ÷ 24 = 8 cm.

Unit conversion in trapezium area

Convert all lengths to the same unit before applying the formula, and express the final area in the corresponding square unit.

For example, 1 m = 100 cm, but 1 m² = 10,000 cm². A length conversion must therefore be squared when converting an already calculated area.

Example: A = 2 m² equals 2 × 10,000 = 20,000 cm².

Relationship with other figures

A parallelogram is a special trapezium when it has two pairs of parallel sides, and its area formula is consistent with the trapezium formula.

If a = b, then A = ½(a + a)h = ah, which is the parallelogram area formula. If one base becomes zero, the expression becomes ½bh, the area formula for a triangle.

Example: For equal parallel sides of 8 cm and height 5 cm, A = ½(8 + 8) × 5 = 40 cm².

Area of Trapezium Visual Guide

Use the diagram or chart below to understand the concept clearly.

Trapezium with labelled bases and height

Visual guide: Trapezium with labelled bases and height
Parallel side a: 14 cm Parallel side b: 8 cm Perpendicular height h: 5 cm
The area uses the two parallel sides and the perpendicular height.

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Practice Area of Trapezium Questions

Practise published questions related to this topic.

1The price of petrol is increased by 28%. A person wants to increase his expenditure by 22% only. By approximately what percent should he decrease his consumption?→ 2An article is sold for Rs. 288 after successive discounts of 20% and 25%. What is the marked price of the article?→ 321.6 ÷ 3.6 × 2 + 0.25 × 16 ÷ 4 - 6 is equal to∶→ 4What is the value of x so that the seven digit number 55350x2 is divisible by 72?→ 5Two articles are sold for Rs. 4,752 each. On one, the seller gains 32% and on the other he loses 28%. What is his overall gain or loss percentage, correct to one decimal place?→ 6The price of sugar is increased by 17%. A person wants to increase his expenditure by 5% only. By approximately what percent should he decrease his consumption?→ 7The value of 15.2 + 5.8 ÷ 2.9 × 2 - 3.5 × 2 ÷ 0.5 is equal to:→ 8What is the value of x so that the seven digit number 8439x53 is divisible by 99?→ 9Two articles are sold for Rs. 962 each. On one, the seller gains 30% and on the other he loses 26%. What is his overall gain or loss percentage, nearest to one decimal place?→ 10An article is sold for Rs. 288 after successive discounts of 25% and x%. If the marked price of the article is Rs. 480, what is the value of x?→

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Quick Revision Notes

Area of Trapezium Revision Points

Remember these formulas, conditions and unit rules.

  • Area = ½ × sum of parallel sides × perpendicular height.
  • The height must be perpendicular to both parallel sides.
  • Average-base form: area = [(a + b) ÷ 2] × h.
  • Height = 2A ÷ (a + b).
  • Unknown parallel side = 2A ÷ h − known parallel side.
  • Use square units such as cm², m² or km² for area.
  • If the two parallel sides are equal, the trapezium formula becomes the parallelogram formula.

Related Topic Explainers

Area of Square → Perimeter of Square → Area of Rectangle → Perimeter of Rectangle → Area of Triangle → Perimeter of Triangle → Heron's Formula → Area of Triangle Using Two Sides and Included Angle →

Area of Trapezium FAQs

What is the area of trapezium formula?

The formula is A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height.

What is the area of a trapezium with parallel sides 6 cm and 10 cm and height 4 cm?

A = ½(6 + 10) × 4 = 32 cm².

How do you find the height when the area is 75 m² and the parallel sides are 8 m and 12 m?

Use h = 2A/(a + b) = 150/20 = 7.5 m.

Can the slanting side be used as the height of a trapezium?

Only if the slanting side is perpendicular to both parallel sides. In a usual oblique trapezium, the height is the perpendicular distance between the bases.

What happens when both parallel sides of a trapezium are equal?

The formula becomes A = ½(a + a)h = ah, which is the area formula for a parallelogram.

How is trapezium area converted from square metres to square centimetres?

Multiply by 10,000 because 1 m² = 10,000 cm². Thus, 3 m² = 30,000 cm².

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