IP Maths Notes (Lower Sec, Year 1-2): 07) Mensuration & Circles
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Mensuration blends geometry with measurement. You must recall formulae, plug numbers correctly, and interpret units.
Learning targets
- Recall core area and circumference formulas for polygons and circles.
- Compute arc length and sector area using radian or degree measures.
- Evaluate surface area and volume of prisms, cylinders, pyramids, and composite solids.
- Justify unit conversions and rounding for dimensional answers.
1. Formula bank
| Figure | Perimeter / Circumference | Area |
| Square | \( 4a \) | \( a^2 \) |
| Rectangle | \( 2(l + b) \) | \( lb \) |
| Triangle | Sum of sides | \( \frac{1}{2}bh \) |
| Circle | \( 2\pi r \) | \( \pi r^2 \) |
For composite shapes, decompose into simpler pieces and add or subtract areas.
2. Circle sectors
- Arc length: \( L = \frac{\theta}{360^\circ} \times 2\pi r \).
- Sector area: \( A = \frac{\theta}{360^\circ} \times \pi r^2 \).
Using radians, replace \( \frac{\theta}{360^\circ} \) with \( \frac{\theta}{2\pi} \).
Worked example: A sector has radius 9 cm and angle \( 40^\circ \).
\[ L = \frac{40}{360} \times 2\pi \times 9 = 2\pi , \text{cm}. \]
\[ A = \frac{40}{360} \times \pi \times 9^2 = 30\pi , \text{cm}^2. \]
3. Surface area and volume
| Solid | Surface area | Volume |
| Prism | Sum of areas of all faces | Base area \(\times\) height |
| Cylinder | \( 2\pi r h + 2\pi r^2 \) | \( \pi r^2 h \) |
| Cone | \( \pi r l + \pi r^2 \) | \( \frac{1}{3}\pi r^2 h \) |
| Sphere | \( 4\pi r^2 \) | \( \frac{4}{3}\pi r^3 \) |
Composite solids
Break the solid into components, compute each measurement, and combine. Keep diagrams annotated with dimensions and units.
Worked example — Cylinder with hemispherical ends
A water bottle consists of a cylinder of radius 3 cm and height 12 cm with two hemispherical ends of the same radius. Find the volume.
Volume = volume of cylinder + volume of sphere (two hemispheres).
\[ V = \pi r^2 h + \frac{4}{3}\pi r^3 = \pi (3)^2 (12) + \frac{4}{3}\pi (3)^3 = 108\pi + 36\pi = 144\pi , \text{cm}^3. \]
4. Units and accuracy
- Use \( \text{cm}^2 \) for area, \( \text{cm}^3 \) for volume.
- Convert units before substituting (e.g., mm to cm).
- Quote answers to 3 significant figures unless the question specifies otherwise.
Try it yourself
- Find the area of a semicircle with radius 7.2 cm.
- Compute the length of an arc in a circle of radius 10 cm subtended by \( 75^\circ \).
- Determine the surface area of a cylinder with radius 4 cm and height 11 cm.
- A solid consists of a cube of side 5 cm with a pyramid of height 6 cm on top (same base). Find the total volume.
Next topic: https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-08-Trigonometry-in-Right-Triangles.
