Revision Guide: Volumes and Surface Areas for Cambridge Lower Secondary Mathematics

The volume of a three-dimensional object is a measure of the space it occupies, typically measured in cubic units such as $\text{cm}^3$ , $\text{m}^3$ , or $\text{ft}^3$ .
General Principle for Prisms: The volume of any prism is calculated by multiplying the area of its constant cross-section (base) by its length (or height).
- Formula: $\text{Volume} = \text{Area of cross-section} \times \text{length}$

A cuboid is a specific type of prism where every face is a rectangle.
Calculation Method:
- The volume is the product of its length ( $l$ ), width ( $w$ ), and height ( $h$ ).
- Formula: $\text{Volume} = l \times w \times h$
Example Case Study (from Page 8):
- Dimensions: Length = $10\,cm$ , Width = $5\,cm$ , Height = $3\,cm$ .
- Step-by-step Calculation:
  - $\text{Volume} = 10\,cm \times 5\,cm \times 3\,cm$
  - $\text{Volume} = 50\,cm^2 \times 3\,cm$
  - $\text{Volume} = 150\,cm^3$

A triangular prism has a cross-section in the shape of a triangle.
Calculation Method:
- First, determine the area of the triangular cross-section: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{vertical height}$ .
- Second, multiply this area by the length of the prism.
Example Case Study (from Page 4):
- Dimensions Provided:
  - Triangle Base ( $b$ ) = $4\,cm$
  - Triangle Vertical Height ( $h$ ) = $3\,cm$
  - Triangle Hypotenuse = $5\,cm$ (Note: This is not used for volume calculation but is relevant for surface area).
  - Prism Length ( $L$ ) = $10\,cm$
- Step-by-step Calculation:
  - $\text{Area of Triangle} = \frac{1}{2} \times 4\,cm \times 3\,cm = 6\,cm^2$
  - $\text{Volume} = 6\,cm^2 \times 10\,cm = 60\,cm^3$

A cylinder is treated as a prism with a circular cross-section.
Calculation Method:
- Area of the circular base is calculated using $\pi \times r^2$ , where $r$ is the radius.
- The volume is then the base area multiplied by the height ( $h$ ).
- Formula: $\text{Volume} = \pi \times r^2 \times h$
Example Case Study (from Page 6):
- Dimensions Provided:
  - Radius ( $r$ ) = $2\,ft$
  - Height ( $h$ ) = $6\,ft$
- Step-by-step Calculation:
  - $\text{Base Area} = \pi \times (2\,ft)^2 = 4\pi \,ft^2$
  - $\text{Volume} = 4\pi \,ft^2 \times 6\,ft = 24\pi \,ft^3$
  - Numerical Approximation: $\text{Volume} \approx 75.40\,ft^3$ (using $\pi \approx 3.14159$ ).

The total surface area ( $SA$ ) of a 3D solid is the sum of the areas of all its exterior faces.
It is measured in square units (e.g., $\text{cm}^2$ , $\text{m}^2$ , $\text{ft}^2$ ).

A cuboid has 6 rectangular faces consisting of 3 pairs of identical rectangles.
Formula:
- $SA = 2 \times (l \times w) + 2 \times (l \times h) + 2 \times (w \times h)$
Example Case Study (from Page 18):
- Dimensions: $10\,cm$ (length), $5\,cm$ (width), $3\,cm$ (height).
- Step-by-step Calculation:
  - Area of top/bottom faces: $2 \times (10 \times 5) = 100\,cm^2$
  - Area of front/back faces: $2 \times (10 \times 3) = 60\,cm^2$
  - Area of side faces: $2 \times (5 \times 3) = 30\,cm^2$
  - $\text{Total Surface Area} = 100 + 60 + 30 = 190\,cm^2$

A triangular prism typically consists of 5 faces: 2 identical triangular bases and 3 rectangular sides.
Example Case Study (from Page 12):
- Dimensions Provided:
  - Triangle base = $4\,cm$ , Triangle vertical height = $3\,cm$ , Triangle hypotenuse = $5\,cm$ .
  - Prism length = $10\,cm$ .
- Step-by-step Calculation:
  - Area of 2 Triangles: $2 \times (\frac{1}{2} \times 4\,cm \times 3\,cm) = 12\,cm^2$
  - Area of Rectangle 1 (bottom): $4\,cm \times 10\,cm = 40\,cm^2$
  - Area of Rectangle 2 (vertical side): $3\,cm \times 10\,cm = 30\,cm^2$
  - Area of Rectangle 3 (slanted side): $5\,cm \times 10\,cm = 50\,cm^2$
  - $\text{Total Surface Area} = 12 + 40 + 30 + 50 = 132\,cm^2$

The surface area of a cylinder consists of two circular bases and one curved surface (which is essentially a rectangle when flattened).
Formula Components:
- Area of two circles: $2 \times \pi \times r^2$
- Area of curved surface: $\text{Circumference} \times \text{height} = 2 \times \pi \times r \times h$
- Total Formula: $SA = 2\pi r^2 + 2\pi rh$
Example Case Study (from Page 14):
- Dimensions: Radius ( $r$ ) = $2\,ft$ , Height ( $h$ ) = $6\,ft$ .
- Step-by-step Calculation:
  - $\text{Two Circles} = 2 \times \pi \times (2)^2 = 8\pi \,ft^2$
  - \text{Curved Surface} = 2 \times \pi \times 2 \times 6 = 24̖\pi \,ft^2
  - $\text{Total Surface Area} = 8\pi + 24\pi = 32\pi \,ft^2$
  - Numerical Approximation: $\text{Total Surface Area} \approx 100.53\,ft^2$

The content focuses on a square-based pyramid.
Calculation Method:
- Find the area of the square base: $\text{side} \times \text{side}$ .
- Find the area of the 4 identical triangular faces: $4 \times (\frac{1}{2} \times \text{base} \times \text{slant height})$ .
- Sum these areas for the total result.

Source Material: This material is derived from the "Cambridge Lower Secondary Mathematics Learner's Book 9" (Second Edition).
Authors: Lynn Byrd, Greg Byrd, and Chris Pearce.
Publisher: Cambridge University Press.
Relevant Section: Practice 17, Questions 3 and 4, located on page 301.
Endorsements: The textbook is endorsed by Cambridge Assessment International Education for full syllabus coverage.