CBSE Class 8 Mathematics Mensuration Notes Set 16

Areas of Plane Figures

Rectangle: area = l x b, perimeter = 2(l + b), diagonal = sqrt(l^2 + b^2). Square: area = a^2 = d^2/2, perimeter = 4a.
Triangle: area = (1/2) x base x height; by Heron's formula area = sqrt(s(s - a)(s - b)(s - c)) with s = (a + b + c)/2. Equilateral triangle of side a: height = (sqrt(3)/2) a.
Parallelogram: base x height. Rhombus: (1/2) x d1 x d2, with diagonals that bisect each other at right angles. Trapezium: (1/2) x (sum of parallel sides) x distance between them.
Circle: circumference = 2 pi r, area = pi r^2. Sector with angle q degrees: area = (q/360) pi r^2.

Solids

Cuboid: volume l b h, total surface area 2(lb + bh + hl), space diagonal sqrt(l^2 + b^2 + h^2). Cube: volume a^3, surface area 6a^2, diagonal a sqrt(3).
Cylinder: volume pi r^2 h, curved surface 2 pi r h, total surface 2 pi r (r + h).
Cone: slant height l = sqrt(r^2 + h^2), volume (1/3) pi r^2 h, curved surface pi r l, total surface pi r (r + l).

Useful Ideas for Word Problems

Wheels: distance covered in one revolution = circumference, so number of revolutions = total distance / circumference.
Paths and borders: area of path = outer area - inner area.
Cost problems: cost = area (or length) x rate; keep units consistent (hectare = 10000 square metres).
Recasting a solid: volume stays the same when a solid is melted and made into another shape.

Worked example: a cone of radius 12 cm and height 50 cm has volume (1/3) pi (144)(50) = 2400 pi cm^3. Poured into a cylinder of radius 10 cm, the liquid rises to 2400 pi / (100 pi) = 24 cm.