Mensuration 2D

Mensuration is the branch of geometry that deals with measurement of lengths, areas, and perimeters of 2D shapes like squares, triangles, circles, and more.

Think of it as the math that answers questions like:
“How much paint is needed to cover this wall?” or “What’s the border length of this football field?”

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Key 2D Shapes, Formulas, and Properties

We'll cover the following standard figures:

• Triangle
• Quadrilaterals (including Square, Rectangle, Rhombus, Parallelogram, Trapezium)
• Circle
• Sector & Segment
• Ellipse
• Composite figures

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1. Triangle

Let the sides be $a, b, c$, base $b$, height $h$, semi-perimeter $s = \frac{a+b+c}{2}$

• Area:

  $$A = \frac{1}{2} \times \text{base} \times \text{height}$$

  or using Heron’s formula:

  $$A = \sqrt{s(s-a)(s-b)(s-c)}$$

• Equilateral triangle:

  $$A = \frac{\sqrt{3}}{4} a^2 \quad ; \quad \text{Height} = \frac{\sqrt{3}}{2} a$$

• Right-angled triangle:

  $$A = \frac{1}{2} \times \text{Perpendicular} \times \text{Base}$$

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2. Square

Side = $a$

• Area: $a^2$
• Perimeter: $4a$
• Diagonal: $a\sqrt{2}$

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3. Rectangle

Length = $l$, Breadth = $b$

• Area: $l \times b$
• Perimeter: $2(l + b)$
• Diagonal: $\sqrt{l^2 + b^2}$

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4. Parallelogram

Base = $b$, Height = $h$

• Area: $b \times h$
• Perimeter: $2(a + b)$, where $a$ and $b$ are adjacent sides
• Diagonals do not bisect at right angles

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5. Rhombus

All sides = $a$, Diagonals = $d_1, d_2$

• Area:

  $$A = \frac{1}{2} d_1 d_2 \quad ; \quad \text{or} \quad a^2 \sin(\theta)$$

• Perimeter: $4a$

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6. Trapezium (Trapezoid)

Parallel sides = $a, b$; height = $h$

• Area:

  $$A = \frac{1}{2}(a + b) \times h$$

• Perimeter: $a + b + \text{non-parallel sides}$

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7. Circle

Radius = $r$, Diameter = $2r$

• Circumference:

  $$C = 2\pi r$$

• Area:

  $$A = \pi r^2$$

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8. Sector of a Circle

Angle subtended = $\theta$ (in degrees), radius = $r$

• Area:

  $$A = \frac{\theta}{360} \times \pi r^2$$

• Arc length:

  $$L = \frac{\theta}{360} \times 2\pi r$$

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9. Segment of a Circle

Area of segment = Area of sector − Area of triangle

• Formula (for minor segment):

  $$A = \frac{\theta}{360} \pi r^2 - \frac{1}{2} r^2 \sin(\theta)$$

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10. Ellipse

Major axis = $2a$, Minor axis = $2b$

• Area:

  $$A = \pi a b$$

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11. Composite Figures

Composite figures are made by combining multiple basic shapes — break them down systematically.

• Total Area = Sum of individual areas
• Border Length = Outer perimeter only
• Avoid double-counting overlapping sections

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Visual Intuition and Derivations

1. Triangle:
   Area as “half the rectangle” it forms when duplicated.

2. Circle sector:
   Sector is just a slice of the whole pizza — proportional to angle $\theta$.

3. Ellipse area:
   Think of stretching a circle in one direction.

4. Heron’s Formula:
   Uses only side lengths when height isn’t known.

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Conceptual Tips and Common Mistakes

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Examples

Example 1

Find the area of a trapezium with parallel sides 10 cm and 14 cm, and height 8 cm.

$$A = \frac{1}{2}(10 + 14) \times 8 = 96 \text{ cm}^2$$

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Example 2

A circle has radius 7 cm. Find area and circumference.

• Area = $\pi r^2 = 22/7 \times 7^2 = 154$ cm²
• Circumference = $2\pi r = 2 \times 22/7 \times 7 = 44$ cm

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Example 3

Find the area of an equilateral triangle of side 6 cm.

$$A = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \approx 15.59 \text{ cm}^2$$

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Example 4

A rhombus has diagonals of 10 cm and 8 cm. Find its area.

$$A = \frac{1}{2} \times 10 \times 8 = 40 \text{ cm}^2$$

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Example 5

Find the area of a sector of radius 14 cm and angle 45°.

$$A = \frac{45}{360} \times \pi r^2 = \frac{1}{8} \times \pi \times 14^2 = \frac{1}{8} \times \pi \times 196 \approx 76.96 \text{ cm}^2$$