Surface Area and Capacity of 3-D Shapes (Continued)
⭐ Higher Tier Content
This topic extends work on three dimensional shapes to include surface area of a cylinder and capacity of curved and pointed solids. These problems often require combining formulas, careful substitution and correct unit conversions.
Surface Area of a Cylinder
A cylinder has two flat circular faces and one curved surface.
The curved surface area is found by multiplying the circumference of the base by the height.
Circumference of the circular base is:
$$
2\pi r
$$
Curved surface area of a cylinder is:
$$
2\pi rh
$$
The total surface area of a cylinder includes the curved surface and the two circular ends.
Area of the two circular faces is:
$$
2\pi r^2
$$
Total surface area of a cylinder is:
$$
2\pi r^2 + 2\pi rh
$$
Surface area is measured in square units such as \( cm^2 \) or \( m^2 \).
Capacity of a Sphere
A sphere is a completely curved solid with no flat faces.
The capacity of a sphere is found using its volume.
Volume of a sphere is:
$$
\frac{4}{3}\pi r^3
$$
The radius must be cubed, so unit handling is especially important.
Capacity is found by converting volume into litres or millilitres when required.
Capacity of a Cone
A cone has a circular base and a curved surface that meets at a point.
The volume of a cone is:
$$
\frac{1}{3}\pi r^2 h
$$
This formula shows that a cone has one third of the volume of a cylinder with the same base and height.
Capacity is again found by converting cubic units into litres or millilitres.
Capacity of a Pyramid
A pyramid has a polygon base and triangular faces that meet at an apex.
The volume of any pyramid is:
$$
\frac{1}{3} \times base\ area \times height
$$
The base area depends on the shape of the base, such as a square or rectangle.
Capacity problems involving pyramids require careful calculation of the base area before using the formula.
Capacity of a Compound Solid
A compound solid is formed by joining two or more simple solids.
To find the capacity of a compound solid:
• split the solid into known shapes
• find the volume of each part
• add the volumes together
If a section is removed, subtract its volume instead.
Only the internal volume is used when finding capacity.
Volume must be converted to capacity using:
$$
1\ cm^3 = 1\ ml
$$
$$
1\ m^3 = 1000\ litres
$$
Using Units Correctly
Higher tier problems often involve mixed units.
Always:
• convert all measurements to the same unit before calculating
• use cubic units for volume
• convert to litres or millilitres only at the final stage
Unit errors are the most common cause of lost marks
Key Points to Remember
The total surface area of a cylinder includes curved and flat surfaces.
The volume of a sphere is \( \frac{4}{3}\pi r^3 \).
Cones and pyramids have one third of the volume of a corresponding prism.
Compound solids must be split into simpler shapes.
Capacity is found by converting volume into litres or millilitres.
Higher tier problems with 3-D shapes require accurate substitution, careful unit handling and clear, structured working.