Geometry & Mensuration
Area, volume, angles, and geometric properties
Angles:
- Angles on a straight line = 180°
- Angles in a triangle = 180°
- Angles in a quadrilateral = 360°
- Vertically opposite angles are equal
- Corresponding angles (parallel lines) are equal (F-angles)
- Alternate angles (parallel lines) are equal (Z-angles)
2D Shapes — Area formulae:
- Rectangle: A = l × w
- Triangle: A = ½bh
- Circle: A = πr², Circumference = 2πr
- Trapezium: A = ½(a + b)h
3D Shapes — Volume formulae:
- Cuboid: V = l × w × h
- Cylinder: V = πr²h
- Cone: V = ⅓πr²h
- Sphere: V = ⁴⁄₃πr³
Surface area: Sum of area of all faces.
Pythagoras' theorem: a² + b² = c² (right-angled triangles, c = hypotenuse)
Trigonometry (SOHCAHTOA):
- sin θ = Opposite/Hypotenuse
- cos θ = Adjacent/Hypotenuse
- tan θ = Opposite/Adjacent
Key Points to Remember
- 1Triangle area = ½bh; Circle area = πr²; Cylinder volume = πr²h
- 2Pythagoras: a² + b² = c²
- 3SOHCAHTOA for trigonometry in right triangles
- 4Corresponding and alternate angles are equal in parallel lines
Pakistan Example
The Minar-e-Pakistan — Geometry in Lahore's Architecture
The Minar-e-Pakistan tower is approximately 60m tall. Standing 80m away, the angle of elevation to the top can be found: tan θ = 60/80 = 0.75 → θ = 36.9°. The circular base has radius 15m → area = π × 15² ≈ 707 m². Pakistan's construction boom (CPEC infrastructure) constantly requires trigonometry and mensuration calculations on site.
Quick Revision Infographic
Mathematics — Quick Revision
Geometry & Mensuration
Key Concepts
Formulas to Know
Circle area = πr²; Cylinder volume = πr²h= c²The Minar-e-Pakistan — Geometry in Lahore's Architecture
The Minar-e-Pakistan tower is approximately 60m tall. Standing 80m away, the angle of elevation to the top can be found: tan θ = 60/80 = 0.75 → θ = 36.9°. The circular base has radius 15m → area = π × 15² ≈ 707 m². Pakistan's construction boom (CPEC infrastructure) constantly requires trigonometry and mensuration calculations on site.