2D and 3D shapes (Areas, Perimeters, Volumes, and Surface Areas).

Mensuration is the measurement of lengths, areas, and volumes. For flat (2D) shapes you find perimeters and areas; for solid (3D) shapes you find surface areas and volumes. Most questions need only the standard formulas and careful handling of units.

2D shapes: perimeter and area

• Rectangle: area = l x b, perimeter = 2(l + b).
• Square: area = a^2, perimeter = 4a.
• Triangle: area = 1/2 x base x height.
• Triangle with sides a, b, c (Heron's formula): area = √(s(s - a)(s - b)(s - c)), where s = (a + b + c) / 2.
• Parallelogram: area = base x height.
• Trapezium: area = 1/2 x (sum of parallel sides) x height.
• Circle: circumference = 2 x pi x r, area = pi x r^2.

3D shapes: surface area and volume

• Cuboid: volume = l x b x h, total surface area = 2(lb + bh + lh).
• Cube: volume = a^3, total surface area = 6a^2.
• Cylinder: volume = pi x r^2 x h, curved surface area = 2 x pi x r x h.
• Cone: volume = 1/3 x pi x r^2 x h, curved surface area = pi x r x l, where the slant height l = √(r^2 + h^2).
• Sphere: volume = 4/3 x pi x r^3, surface area = 4 x pi x r^2.

Points to remember

• Lengths are in cm or m, areas in cm^2 or m^2, and volumes in cm^3 or m^3.
• Convert all dimensions to the same unit before calculating.
• Use pi = 22/7 when the radius is a multiple of 7, and 3.14 otherwise, unless a value is given.
• When a solid is melted and recast, its volume stays the same.

How to solve questions

1. Identify the shape and what is asked: perimeter, area, surface area, or volume.
2. Write the formula and the given dimensions in the same unit.
3. Substitute carefully and calculate.
4. State the answer with the correct unit.