Mensuration, Pythagoras and Trigonometry

Mensuration is the measurement of geometric figures — their lengths, perimeters, areas, surface areas and volumes. Pythagoras' theorem and trigonometry extend this to right-angled triangles.

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Perimeter and Area of 2D Shapes

Shape	Perimeter	Area
Rectangle	2(l + w)	l × w
Triangle	a + b + c	(1/2) × b × h
Parallelogram	2(a + b)	b × h (perpendicular height)
Trapezium	a + b + c + d	(1/2)(a + b) × h
Circle	2πr = πd	πr²

π ≈ 3.14159… — leave answers in terms of π unless asked for a decimal.

Example: Area of a trapezium with parallel sides 8 cm and 12 cm, height 5 cm: = (1/2)(8 + 12) × 5 = (1/2) × 20 × 5 = 50 cm²

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Circles

Circumference: C = 2πr = πd Area: A = πr²

Example: A circle has radius 7 cm. Circumference = 2 × π × 7 = 14π ≈ 43.98 cm Area = π × 49 = 49π ≈ 153.94 cm²

Arc Length and Sector Area

For a sector with angle θ° and radius r:

• Arc length = (θ/360) × 2πr
• Sector area = (θ/360) × πr²

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Composite Shapes

Break the shape into standard parts, calculate each area, then add (or subtract for holes).

Example: An L-shaped room measuring 10 m × 6 m with a 4 m × 3 m rectangular notch removed. Full rectangle area = 60 m²; notch = 12 m²; room area = 48 m²

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Volume and Surface Area of 3D Shapes

Shape	Volume	Surface Area
Cuboid	l × w × h	2(lw + lh + wh)
Prism	Area of cross-section × length	2 × base area + perimeter × length
Cylinder	πr²h	2πr² + 2πrh
Cone	(1/3)πr²h	πr² + πrl (l = slant height)
Sphere	(4/3)πr³	4πr²

Example: Volume of a cylinder, r = 5 cm, h = 12 cm: V = π × 25 × 12 = 300π ≈ 942.5 cm³

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Pythagoras' Theorem

In any right-angled triangle, where c is the hypotenuse (the longest side, opposite the right angle):

a² + b² = c²

Finding the hypotenuse: a = 5, b = 12 → c² = 25 + 144 = 169 → c = 13