Mensuration Formulas: Area and Volume
Mensuration is the branch of mathematics that deals with measuring the area, perimeter, surface area and volume of shapes. It carries good marks in CBSE and ICSE board exams, especially in Class 9 and 10. This article collects the most important formulas and shows a worked example so you can apply them confidently.
2D shapes: area and perimeter
Two-dimensional (flat) shapes have area, measured in square units, and perimeter, the distance around the boundary.
- Square: area = side²; perimeter = 4 × side.
- Rectangle: area = length × breadth; perimeter = 2(length + breadth).
- Triangle: area = ½ × base × height.
- Circle: area = πr²; circumference = 2πr.
- Parallelogram: area = base × height.
For a triangle with all three sides known but no height, use Heron's formula: area = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2.
3D shapes: surface area and volume
Three-dimensional (solid) shapes have surface area, measured in square units, and volume, measured in cubic units.
- Cube: volume = a³; total surface area = 6a².
- Cuboid: volume = l × b × h; total surface area = 2(lb + bh + hl).
- Cylinder: volume = πr²h; total surface area = 2πr(r + h).
- Cone: volume = ⅓πr²h; total surface area = πr(r + l), where l is the slant height.
- Sphere: volume = (4/3)πr³; surface area = 4πr².
Note that the slant height of a cone is found using the Pythagoras theorem: l = √(r² + h²).
Worked example
A cylindrical water tank has radius 7 m and height 10 m. Find its volume and total surface area. Take π = 22/7.
Volume = πr²h = (22/7) × 7 × 7 × 10 = 22 × 7 × 10 = 1540 cubic metres.
Total surface area = 2πr(r + h) = 2 × (22/7) × 7 × (7 + 10) = 2 × 22 × 17 = 748 square metres.
So the tank holds 1540 m³ of water and has a surface area of 748 m².
Tips for board exams
Always write the formula before substituting values, keep units consistent, and remember whether the question asks for curved surface area or total surface area. Practising these regularly is key, as covered in Class 9 Maths and continued in Class 10 Maths.
Frequently asked questions
What is the difference between curved and total surface area?
Curved surface area covers only the curved (side) part of a solid like a cylinder or cone. Total surface area adds the flat circular ends as well. Read the question carefully to know which one is needed.
Which value of pi should I use in exams?
Use π = 22/7 when the numbers are multiples of 7, as the arithmetic stays neat. Otherwise use π = 3.14, unless the question states a specific value.
Why are area units squared and volume units cubed?
Area measures a flat 2D region, so it multiplies two lengths, giving square units. Volume measures 3D space, multiplying three lengths, giving cubic units.