Mensuration: Plane Figures and Solid Shapes

Mensuration is the branch of mathematics that deals with the measurement of lengths, areas, and volumes of geometric figures. In the JAMB syllabus, this covers plane shapes, circles, 3D solids, and Earth geometry.

1. Plane Geometrical Figures

• Perimeter: The total distance around the edge of a shape.
• Area: The amount of space inside a 2D shape.
• Triangle: Area = 1/2 × base × height OR Heron’s Formula: √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2.
• Trapezium: Area = 1/2(a + b)h, where a and b are parallel sides.
• Rhombus/Kite: Area = 1/2 × product of diagonals (d1 × d2).

2. The Circle: Arcs, Sectors, and Segments

• Arc Length (L): (θ/360) × 2πr
• Perimeter of Sector: Arc length + 2r
• Area of Sector: (θ/360) × πr²
• Chord Length: 2r sin(θ/2)
• Area of Segment: Area of Sector - Area of Triangle = (θ/360 × πr²) - (1/2 r² sinθ)

3. Surface Areas and Volumes of Solids

• Cuboid: Volume = lwh; Total Surface Area (TSA) = 2(lw + lh + wh).
• Cylinder: Volume = πr²h; Curved Surface Area (CSA) = 2πrh; TSA = 2πr(r + h).
• Cone: Volume = 1/3 πr²h; CSA = πrl (where l is slant height = √(r² + h²)); TSA = πr(r + l).
• Sphere: Volume = 4/3 πr³; Surface Area = 4πr².
• Pyramid: Volume = 1/3 × Base Area × perpendicular height.
• Prism: Volume = Base Area × length.

4. Earth as a Sphere

The Earth is treated as a sphere of radius R (approx. 6400km). Points are defined by Latitude (North/South) and Longitude (East/West).

• Distance along Great Circle (Longitudes or Equator): D = (θ/360) × 2πR.
• Distance along Parallel of Latitude: d = (α/360) × 2πr, where r = R cos Φ (Φ is the latitude).