Measuring flat shapes and solids by area, surface and volume
Mensuration is the measurement of length, area and volume. For plane figures, the area of a triangle is ½ × base × height, or √[s(s − a)(s − b)(s − c)] by Heron's formula when only the three sides are known, where s is half the perimeter. A circle of radius r has circumference 2πr and area πr². A sector with angle θ at the centre has area (θ/360°) × πr², which is useful for questions about clock hands, fans and arcs.
For solids, the curved surface area of a cylinder is 2πrh and its volume is πr²h. A cone with the same base and height has exactly one third of that volume, ⅓πr²h, and its curved surface is πrl, where l is the slant height √(r² + h²). A sphere has surface area 4πr² and volume (4/3)πr³. In combined-solid questions, such as a cone mounted on a hemisphere, add only the surfaces that are visible on the outside, while the volumes simply add. When a solid is melted and recast into another shape, its volume stays the same.
Why it matters
Mensuration answers practical questions: how much paint covers a wall, how much water a tank holds, or how much metal is needed to make a batch of cans. Remember that 1 litre equals 1000 cm³ and 1 m³ equals 1000 litres. These formulas are also used in physics and chemistry calculations, and integration in Class 12 explains where many of them come from.
Key terms
- Heron's formula
- A way to find a triangle's area from its three side lengths using the semi-perimeter.
- Slant height
- The distance from the tip of a cone to any point on the edge of its base.
- Curved surface area
- The area of only the curved part of a solid, excluding its flat ends.
- Sector
- The region of a circle enclosed by two radii and the arc between them.