Mensuration – Quick Study Guide
(For the Duolingo English Test – Quantitative Aptitude)
1. Simple Explanation
Mensuration is the branch of mathematics that deals with finding lengths, areas, and volumes of geometric figures.
- Length → Linear dimensions (e.g., perimeter, circumference).
- Area → Surface coverage of a 2‑D shape.
- Volume → Space occupied by a 3‑D object.
Why it matters for the test
- Questions often ask you to compute an area, volume, or surface area using a short formula.
- Being able to recognize the shape and recall its formula is the fastest route to the answer.
2. Key Points & Formula Sheet
| Shape | 2‑D Formula | 3‑D Formula (if applicable) | Notes |
|---|---|---|---|
| Rectangle | A = l × w |
N/A | Perimeter P = 2(l + w) |
| Square | A = a² |
N/A | Perimeter P = 4a |
| Triangle | A = ½ × b × h (base × height) |
Tetrahedron (regular) V = a³ / (6√2) |
For right triangle, c² = a² + b² |
| Circle | A = πr² |
Sphere V = 4/3 πr³ |
Circumference C = 2πr |
| Parallelogram | A = b × h |
N/A | Same as rectangle if h = w |
| Trapezium (Trapezoid) | A = ½ (a + b) × h |
N/A | a & b are parallel sides |
| Ellipse | A = πab |
N/A | a = semi‑major, b = semi‑minor |
| Cylinder | A_surf = 2πr(h + r) |
V = πr²h |
Lateral surface A_lat = 2πrh |
| Cone | A_surf = πr(r + s) (lateral) |
V = 1/3 πr²h |
s = slant height (s² = r² + h²) |
| Pyramid (regular) | A_surf = B + ½ P × l |
V = 1/3 B h |
B = base area, P = perimeter of base, l = slant height |
| Prism (regular) | A_surf = 2B + P × h |
V = B h |
B = area of one base, P = perimeter of base |
Quick‑Reference List
- π ≈ 3.1416
- √ (square root)
- All formulas assume units are consistent (cm, m, etc.).
3. Worked Example
Problem
A right circular cylinder has a radius of 4 cm and a height of 10 cm.
Tasks
- Find the surface area.
- Find the volume.
Solution
1. Surface Area
r = 4 cm h = 10 cm
A_surf = 2πr(h + r)
= 2 × π × 4 × (10 + 4)
= 8π × 14
= 112π
Numerically: 112 × 3.1416 ≈ 351.86 cm².
2. Volume
V = πr²h
= π × (4)² × 10
= π × 16 × 10
= 160π
Numerically: 160 × 3.1416 ≈ 502.65 cm³.
Answer
- Surface area ≈ 351.9 cm²
- Volume ≈ 502.7 cm³
4. Quick Revision Summary
- Identify the figure → choose the right formula.
- Keep π and √ handy (π ≈ 3.1416, √2 ≈ 1.414).
- Units matter – always use the same unit throughout.
- Common shortcuts
- Area of circle →
πr². - Area of triangle →
½ × base × height. - Volume of cube →
a³. - Volume of sphere →
4/3 πr³. - Surface area of cylinder →
2πr(h + r). - Volume of cone →
1/3 πr²h.
- Area of circle →
- Remember: For any 3‑D shape, volume is always area of base × height / (some factor).
- Practice: Work through at least 5 problems per shape before the test.
Final Tip
When you see a question, draw a quick sketch (even mentally).
- Label known dimensions (radius, height, base, etc.).
- Identify if the shape is right‑angled (e.g., right triangle, right circular cylinder).
- Pick the formula that matches the labeled dimensions.
Good luck!