IB Mathematics AI – Topic 3

Rectangular Prism (Cuboid):

Dimensions: length (l), width (w), height (h)

Volume:

\[ V = l \times w \times h \]

Surface Area:

\[ SA = 2(lw + lh + wh) \]

Sum of areas of all 6 rectangular faces

Special Case - Cube:

When l = w = h = s (side length):

\[ V = s^3 \]

\[ SA = 6s^2 \]

Cylindrical Prism (Cylinder):

Dimensions: radius (r), height (h)

Volume:

\[ V = \pi r^2 h \]

Base area × height

Surface Area:

\[ SA = 2\pi r^2 + 2\pi rh \]

Two circular bases + curved surface area

Or: \(SA = 2\pi r(r + h)\)

⚠️ Common Pitfalls & Tips:

• Don't confuse diameter with radius (r = d/2)
• Surface area includes ALL faces (top, bottom, and sides)
• Units: Volume is always cubed (cm³, m³)
• Units: Surface area is always squared (cm², m²)

General Pyramid Formula:

\[ V = \frac{1}{3} \times \text{Base Area} \times h \]

where h is the perpendicular height from base to apex

Common Types:

1. Square-based Pyramid:

Base is a square with side length s

\[ V = \frac{1}{3}s^2h \]

2. Rectangular-based Pyramid:

Base has dimensions l × w

\[ V = \frac{1}{3}lwh \]

Surface Area:

Sum of base area and all triangular faces

For each triangular face: \(A = \frac{1}{2} \times \text{base} \times \text{slant height}\)

⚠️ Common Pitfalls & Tips:

• Key factor: 1/3 – pyramids are 1/3 the volume of corresponding prism
• Use perpendicular height (h), not slant height, for volume
• Slant height is used for surface area of faces

Key Dimensions:

• r: radius of circular base
• h: perpendicular height
• l: slant height

Relationship (Pythagoras):

\[ l^2 = r^2 + h^2 \]

Volume:

\[ V = \frac{1}{3}\pi r^2 h \]

Curved Surface Area:

\[ A_{\text{curved}} = \pi r l \]

where l is the slant height

Total Surface Area:

\[ SA = \pi r^2 + \pi r l = \pi r(r + l) \]

Circular base + curved surface

⚠️ Common Pitfalls & Tips:

• Volume uses perpendicular height (h), not slant height (l)
• Curved surface area uses slant height (l)
• If given diameter, convert to radius first
• Use Pythagoras if you need to find missing dimension

Solution:

Given: r = 6 cm, l = 10 cm

(a) Find perpendicular height h:

Using Pythagoras: \(l^2 = r^2 + h^2\)

\[ h^2 = l^2 - r^2 = 10^2 - 6^2 = 100 - 36 = 64 \]

\[ h = \sqrt{64} = 8 \text{ cm} \]

(b) Calculate volume:

Using \(V = \frac{1}{3}\pi r^2 h\)

\[ V = \frac{1}{3}\pi (6)^2(8) = \frac{1}{3}\pi (36)(8) = 96\pi \]

\[ V \approx 301.6 \text{ cm}^3 \]

(c) Calculate total surface area:

Using \(SA = \pi r(r + l)\)

\[ SA = \pi (6)(6 + 10) = \pi (6)(16) = 96\pi \]

\[ SA \approx 301.6 \text{ cm}^2 \]

Answers: (a) 8 cm, (b) 302 cm³, (c) 302 cm²

Sphere:

A perfectly round 3D shape where all points on the surface are equidistant from the center

Volume of Sphere:

\[ V = \frac{4}{3}\pi r^3 \]

Surface Area of Sphere:

\[ SA = 4\pi r^2 \]

Hemisphere (Half Sphere):

A hemisphere is exactly half of a sphere

Volume of Hemisphere:

\[ V = \frac{2}{3}\pi r^3 \]

(Half of sphere volume)

Surface Area of Hemisphere:

\[ SA = 3\pi r^2 \]

Curved surface (\(2\pi r^2\)) + circular base (\(\pi r^2\))

⚠️ Common Pitfalls & Tips:

• Sphere formulas have fractions: 4/3 for volume, not 1/3
• Hemisphere volume is 2/3πr³, not 1/3πr³
• Hemisphere SA includes the flat circular base
• Always cube the radius for volume calculations

Solution:

Given: r = 5 cm, cylinder height h = 12 cm

(a) Total Volume:

Volume of cylinder:

\[ V_{\text{cyl}} = \pi r^2 h = \pi (5)^2(12) = 300\pi \text{ cm}^3 \]

Volume of hemisphere:

\[ V_{\text{hem}} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi (5)^3 = \frac{2}{3}\pi (125) = \frac{250}{3}\pi \text{ cm}^3 \]

Total volume:

\[ V_{\text{total}} = 300\pi + \frac{250}{3}\pi = \frac{900\pi + 250\pi}{3} = \frac{1150}{3}\pi \]

\[ V \approx 1204 \text{ cm}^3 \]

(b) Total Surface Area:

Curved surface of hemisphere:

\[ A_{\text{hem}} = 2\pi r^2 = 2\pi (5)^2 = 50\pi \text{ cm}^2 \]

Curved surface of cylinder:

\[ A_{\text{cyl curved}} = 2\pi rh = 2\pi (5)(12) = 120\pi \text{ cm}^2 \]

Base of cylinder:

\[ A_{\text{base}} = \pi r^2 = \pi (5)^2 = 25\pi \text{ cm}^2 \]

(Note: Top of cylinder is covered by hemisphere, so not included)

Total surface area:

\[ SA = 50\pi + 120\pi + 25\pi = 195\pi \approx 613 \text{ cm}^2 \]

Answers: (a) 1200 cm³, (b) 613 cm²

Units

• Volume: Always cubed (cm³, m³)
• Surface Area: Always squared (cm², m²)
• Match units in question

Key Fractions

• Pyramid/Cone: 1/3
• Sphere: 4/3
• Hemisphere: 2/3

Common Errors

• Diameter vs radius
• Height vs slant height
• Forgetting π in formulas

Composite Shapes

• Break into simple shapes
• Calculate each separately
• Add/subtract as needed