Area Of Parallelogram, Triangles & Circles

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Unit: Geometry

Chapter: Area Of Parallelogram, Triangles & Circles

Reference: – What is Area, Area of a Parallelogram (Formula and Derivation), Area of a Trapezium (Formula and Derivation), Finding Base or Height from Area, Real-Life Applications, Solved Examples, Odd-One-Out Problems, Common Mistakes

After studying this chapter, you should be able to understand:

• How to Find the Area of a Parallelogram
• How to Find the Area of a Trapezium (Trapezoid)
• How to Find Missing Dimensions Given Area
• Difference Between Area Formulas of Different Shapes

Introduction to Area of Trapezium & Parallelogram

Definition

Area is the amount of space enclosed within a two-dimensional shape, measured in square units (cm², m², in², etc.). A parallelogram is a quadrilateral with two pairs of parallel sides. A trapezium (called trapezoid in the US) is a quadrilateral with exactly one pair of parallel sides.

When we calculate area, we essentially ask:

"How many square units can fit inside this shape?"

Once we know the formulas, we can find area quickly without counting squares.

Importance of Area

• Used in construction (flooring, painting, roofing)
• Essential for land measurement and agriculture
• Helps in designing and manufacturing products
• Foundation for volume and surface area in higher grades

Example

The area of a parallelogram with base 10 cm and height 5 cm is 50 cm². The area of a trapezium with bases 8 cm and 12 cm and height 4 cm is 40 cm².

Subtopics

1. Area of a Parallelogram

A parallelogram is a quadrilateral with opposite sides parallel and equal.

Formula: Area = base × height (A = b × h)

Important: The height (h) is the perpendicular distance between the bases, NOT the length of the slanted side.

Derivation: A parallelogram can be transformed into a rectangle by cutting off a right triangle from one end and moving it to the other end. The rectangle has length = base and width = height, so area = b × h.

Example 1: Parallelogram with base 8 cm and height 5 cm
Area = 8 × 5 = 40 cm²

Example 2: Parallelogram with base 12 m and height 7 m
Area = 12 × 7 = 84 m²

Finding Base or Height: If area is known, base = Area ÷ height, height = Area ÷ base

Example – Find height: Area = 45 cm², base = 9 cm → height = 45 ÷ 9 = 5 cm

2. Area of a Trapezium (Trapezoid)

A trapezium is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases (a and b). The perpendicular distance between them is the height (h).

Formula: Area = (1/2) × (sum of parallel sides) × height

A = (1/2) × (a + b) × h

Derivation: A trapezium can be divided into a parallelogram and a triangle, or two triangles, or seen as half of a parallelogram with the same height and base equal to (a + b).

Example 1: Bases = 6 cm and 10 cm, height = 4 cm
Area = (1/2) × (6 + 10) × 4 = (1/2) × 16 × 4 = 8 × 4 = 32 cm²

Example 2: Bases = 5 m and 9 m, height = 6 m
Area = (1/2) × (5 + 9) × 6 = (1/2) × 14 × 6 = 7 × 6 = 42 m²

Finding a Base or Height: If area is known, (a + b) = (2 × Area) ÷ h, or h = (2 × Area) ÷ (a + b)

Example – Find missing base: Area = 50 cm², one base = 8 cm, height = 5 cm
(a + b) = (2 × 50) ÷ 5 = 100 ÷ 5 = 20 → b = 20 – 8 = 12 cm

Solved Examples

Example 1 – Area of Parallelogram: Find the area of a parallelogram with base 15 cm and height 8 cm.

Solution: A = 15 × 8 = 120 cm²

Answer: 120 cm²

Example 2 – Height of Parallelogram: A parallelogram has area 72 m² and base 12 m. Find its height.

Solution: height = Area ÷ base = 72 ÷ 12 = 6 m

Answer: 6 m

Example 3 – Area of Trapezium: Find the area of a trapezium with parallel sides 7 cm and 13 cm and height 5 cm.

Solution: A = (1/2) × (7 + 13) × 5 = (1/2) × 20 × 5 = 10 × 5 = 50 cm²

Answer: 50 cm²

Example 4 – Missing Base of Trapezium: A trapezium has area 84 m², height 7 m, and one base 10 m. Find the other base.

Solution: (a + b) = (2 × 84) ÷ 7 = 168 ÷ 7 = 24 → b = 24 – 10 = 14 m

Answer: 14 m

Common Mistakes to Avoid

Mistake 1 – Using the slanted side as height in a parallelogram
The height must be perpendicular to the base, not the length of the slanted side.
Correct understanding: height is the shortest distance between the bases.

Mistake 2 – Forgetting the 1/2 in trapezium formula
Area = (1/2)(a+b)h, not (a+b)h. Without 1/2, the area would be twice the correct value.
Correct understanding: Always include the 1/2 factor for trapezium.

Mistake 3 – Confusing the two bases in trapezium
The two parallel sides are both bases. The order does not matter because addition is commutative.
Correct understanding: a + b = b + a, so no need to worry which base is which.

Mistake 4 – Using the legs of trapezium as height
The legs (non-parallel sides) are not the height unless perpendicular to the bases.
Correct understanding: Height is the perpendicular distance between the parallel sides.

Mistake 5 – Mixing up trapezium and parallelogram formulas
Parallelogram: A = b × h (no 1/2), Trapezium: A = (1/2)(a+b)h (has 1/2 and two bases).
Correct understanding: Memorize both formulas and know when to use each.

Mistake 6 – Forgetting to use consistent units
If base is in meters and height in centimeters, the area will be incorrect.
Correct understanding: Convert both dimensions to the same unit before multiplying.

Quick Reference Summary

Parallelogram Area: A = b × h
b = length of base, h = perpendicular height

Trapezium Area: A = (1/2) × (a + b) × h
a and b = lengths of parallel sides, h = perpendicular height

Parallelogram – Opposite sides parallel and equal, height perpendicular to base

Trapezium – Exactly one pair of parallel sides, height perpendicular distance between them

To Find Missing Dimension:
Parallelogram: b = A ÷ h, h = A ÷ b
Trapezium: (a+b) = 2A ÷ h, h = 2A ÷ (a+b)

Units: Area is always in square units (cm², m², in², ft², etc.)