Mensuration Study Notes
(For the Quantitative Aptitude section of the Duolingo English Test)
1. What is Mensuration?
Mensuration is simply measuring the size of shapes.
Think of it as finding out how much space a shape covers or how much boundary it has.
2. 2‑D Shapes
| Shape | What to Find | Everyday Analogy | Formula (plain words) | Tiny Example |
|---|---|---|---|---|
| Rectangle | Area – how much floor inside Perimeter – length of the border |
A garden you need to fill with grass | Area = length × width Perimeter = 2 × (length + width) |
Garden 10 m × 5 m → Area 50 m², Perimeter 30 m |
| Square | Same as rectangle | A picture frame with equal sides | Area = side² Perimeter = 4 × side |
Frame side 4 m → Area 16 m², Perimeter 16 m |
| Triangle | Area – how much floor inside Perimeter – length of the border |
A slice of pizza | Area = ½ × base × height Perimeter = sum of all three sides |
Triangle base 6 m, height 4 m → Area 12 m² |
| Circle | Area – how much floor inside Circumference – length of the border |
A round pizza or a round table | Area = π × radius² Circumference = 2 × π × radius |
Radius 3 m → Area ≈ 28.27 m², Circumference ≈ 18.85 m |
Tip – Remember “π” ≈ 3.14.
Shortcut – If you know the diameter (twice the radius), just use Diameter ÷ 2 = radius.
3. 3‑D Shapes
| Shape | Surface Area (how much paint you need) | Volume (how much water it can hold) | Everyday Analogy | Tiny Example |
|---|---|---|---|---|
| Cube | 6 × side² | side³ | A small box of chocolates | Side 2 m → Surface 24 m², Volume 8 m³ |
| Rectangular Prism (box) | 2 × (lw + lh + wh) | l × w × h | A shipping crate | Length 4 m, width 3 m, height 2 m → Surface 52 m², Volume 24 m³ |
| Cylinder | 2 × π × r × (r + h) | π × r² × h | A water bottle | Radius 1 m, height 4 m → Surface ≈ 50.27 m², Volume ≈ 12.57 m³ |
| Cone | π × r × (r + slant height) | ⅓ × π × r² × h | A party hat | Radius 1 m, height 3 m → Surface ≈ 29.61 m², Volume ≈ 3.14 m³ |
| Sphere | 4 × π × r² | ⅔ × π × r³ | A basketball | Radius 0.5 m → Surface ≈ 3.14 m², Volume ≈ 0.52 m³ |
Quick checks
– Volume is “how much space inside”.
– Surface area is “how much paint or coverage needed”.
4. How to Solve a Mensuration Problem
- Identify the shape – Look at the picture or description.
- Know what you need – Area, perimeter, surface area, or volume.
- Grab the right formula – Use the table above.
- Plug in the numbers – Keep units the same (meters, centimeters, etc.).
- Do the arithmetic – Use a calculator if needed.
- Check the answer – Does it make sense? (e.g., volume should be positive, area larger than zero).
5. Mini Practice Problems
| Problem | What to find | Your steps |
|---|---|---|
| 1. A rectangular room is 12 m long and 7 m wide. How many square meters of paint are needed for the walls if the walls are 3 m high? | Surface area of the walls | 2 × (12 m + 7 m) × 3 m = 90 m² |