Geometry and mensuration
Mensuration is the measuring of lengths, areas and volumes of standard shapes. Most questions come down to choosing the right formula, using consistent units, and sometimes splitting a figure into simple pieces. Area is measured in square units and volume in cubic units, so a conversion factor gets squared or cubed.
Formulas and when they apply
- Rectangle: area = l × b, perimeter = 2(l + b). Square: area = a², perimeter = 4a
- Triangle: area = ½ × base × height. With sides a, b, c and s = (a + b + c)/2, area = √[s(s − a)(s − b)(s − c)]Applies when: second form is Heron's formula, for any triangle
- Right triangle: a² + b² = c², where c is the hypotenuseApplies when: right-angled triangles only
- Circle: area = πr², circumference = 2πrApplies when: use π = 22/7 or 3.14 only as the question says
- Cuboid: volume = l × b × h, total surface area = 2(lb + bh + hl). Cube: volume = a³, surface area = 6a²
- Cylinder: volume = πr²h, curved surface = 2πrh, total surface = 2πr(r + h)
- Cone: volume = ⅓πr²h, curved surface = πrl where l = √(r² + h²). Sphere: volume = 4/3 πr³, surface = 4πr²
Worked examples
Example 1. A cone has base radius 7 cm and height 24 cm. Take π = 22/7. Find its curved surface area.
Solution. Slant height l = √(7² + 24²) = √(49 + 576) = √625 = 25 cm. Curved surface area = π r l = 22/7 × 7 × 25 = 22 × 25 = 550 cm².
Example 2. The perimeter of a rectangle is 56 m and its length and breadth are in the ratio 4 : 3. Find its area.
Solution. Length + breadth = 56 ÷ 2 = 28 m. Split 28 in the ratio 4 : 3 into 16 m and 12 m. Area = 16 × 12 = 192 m².
Example 3. A cuboid tank is 5 m long and 4 m wide. How many litres of water are in it when the water is 2.5 m deep? (1 m³ = 1,000 litres)
Solution. Volume of water = 5 × 4 × 2.5 = 50 m³. In litres this is 50 × 1,000 = 50,000 litres.
Shortcuts and their limits
- Remember the right-triangle triples (3, 4, 5), (5, 12, 13), (8, 15, 17) and (7, 24, 25), and their multiples.Does not apply when: Only for right-angled triangles. Check that the largest side is the hypotenuse.
- If all lengths of a shape are multiplied by k, the area is multiplied by k² and the volume by k³.Does not apply when: Only for similar shapes (an exact enlargement).
- When the radius is a multiple of 7, use π = 22/7 so the 7 cancels.Does not apply when: Only when the question allows π = 22/7. Otherwise use the value the question gives.
- A path of width w around a rectangle has area perimeter × w + 4w² outside, or perimeter × w − 4w² inside.Does not apply when: Only for a uniform path around a rectangle, not a circle or an uneven path.
Common mistakes
- Using the diameter where the formula needs the radius.
- Mixing units, such as metres for one side and centimetres for another, or forgetting to convert m³ to litres.
- Using the slant height where the vertical height is needed (cone volume), or the other way round (curved surface).
- Treating a triangle as right-angled when nothing says so.
- Counting the four corner squares twice when finding the area of a path around a rectangle.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.