Surface Area and Volume

The One Thing

Surface area measures the exposed outer area of a solid, whereas volume measures the space enclosed by it. Successful solutions require identifying the solid, selecting the appropriate formula, using consistent units, and including only the surfaces relevant to the question.

Definitions and Results

• Surface Area: The total area of all exposed surfaces of a three-dimensional solid, measured in square units.
• Curved Surface Area: The area of only the curved part of a solid, excluding circular or plane bases where applicable.
• Total Surface Area: The sum of the areas of all surfaces, including curved surfaces and bases.
• Volume: The amount of space occupied by a solid, measured in cubic units.
• Cuboid: A solid with six rectangular faces, described by length, breadth, and height.
• Cube: A cuboid with six equal square faces and all edges of the same length.
• Cylinder: A solid with two equal, parallel circular bases joined by a curved surface.
• Cone: A solid with a circular base and one vertex, joined by a curved surface.
• Sphere: A perfectly round solid in which every point on the surface is the same distance from the centre.
• Hemisphere: Half of a sphere, formed by cutting a sphere through its centre.
• Radius: The distance from the centre of a circle or sphere to its boundary.
• Diameter: The distance across a circle or sphere through its centre; $d=2r$.
• Slant Height: The inclined length of a cone from its vertex to a point on the circular boundary.
• Pi: The ratio of a circle’s circumference to its diameter, commonly taken as $\frac{22}{7}$ or $3.14$.
• Composite Solid: A solid formed by joining two or more simpler solids, such as a cylinder joined to a hemisphere.
• Unit Conversion: Changing measurements into equivalent units, such as centimetres to metres or cubic centimetres to litres.

• Cuboid formulas: For length $l$, breadth $b$, and height $h$:

$$\text{TSA}=2(lb+bh+hl),\text{LSA}=2h(l+b),V=lbh.$$

• Cube formulas: For edge $a$:

$$\text{TSA}=6a^2,\text{LSA}=4a^2,V=a^3.$$

• Cylinder formulas: For radius $r$ and height $h$:

$$\text{CSA}=2\pi rh,\text{TSA}=2\pi r(h+r),V=\pi r^{2}h.$$

• Cone formulas: For radius $r$, height $h$, and slant height $l$:

$$\text{CSA}=\pi rl,\text{TSA}=\pi r(l+r),V=\frac{1}{3}\pi r^{2}h.$$

• Right circular cone relationship:

$$l^2=r^2+h^2.$$

• Sphere formulas: For radius $r$:

$$\text{Surface area}=4\pi r^2,V=\frac{4}{3}\pi r^3.$$

• Hemisphere formulas: For radius $r$:

$$\text{CSA}=2\pi r^2,\text{TSA}=3\pi r^2,V=\frac{2}{3}\pi r^3.$$

• Melted and recast solids: The volume remains unchanged if no material is lost.
• Composite-solid volume: Add the volumes of joined parts or subtract the volumes of removed parts.
• Composite-solid surface area: Count only exposed surfaces; internally joined surfaces are excluded.
• Units: Surface area is measured in square units, such as ${\text{cm}}^2$ or ${\text{m}}^2$. Volume is measured in cubic units, such as ${\text{cm}}^3$ or ${\text{m}}^3$.
• Capacity conversions:

$$1\text{ litre}=1000{\text{ cm}}^3,1\text{ millilitre}=1{\text{ cm}}^3.$$

• Useful conversions:

$$1\text{ m}=100\text{ cm},1{\text{ m}}^2=10,000{\text{ cm}}^2,1{\text{ m}}^3=1,000,000{\text{ cm}}^3.$$

• Dimensional checks: Surface-area formulas must produce units of length squared; volume formulas must produce units of length cubed.

Worked Methods

Finding the surface area or volume of a standard solid

• Identify the solid: cuboid, cube, cylinder, cone, sphere, or hemisphere.
• Draw or inspect a diagram and label all dimensions.
• Determine whether the question requires curved surface area, total surface area, lateral surface area, or volume.
• Convert all measurements into the same unit.
• Substitute the dimensions into the appropriate formula.
• Check that the resulting units are square units for area or cubic units for volume.
• Assess whether the numerical result is reasonable.

For example, a cylinder requires different formulas depending on the requested quantity: $$\text{CSA}=2\pi rh,\text{TSA}=2\pi r(h+r),V=\pi r^{2}h.$$ The bases are excluded from curved surface area but included in total surface area.

Finding a cone’s missing slant height

• Identify the radius $r$ and perpendicular height $h$.
• Use the right circular cone relationship:

$$l^2=r^2+h^2.$$

• Take the positive square root:

$$l=\sqrt{r^2+h^2}.$$

• Use $l$ in the cone surface-area formulas:

$$\text{CSA}=\pi rl,\text{TSA}=\pi r(l+r).$$

Solving a composite-solid problem

• Separate the solid into recognisable simpler solids, such as a cylinder joined to a hemisphere.
• Calculate the volume of each component.
• Add volumes for joined parts:

$$V_{\text{total}}=V_1+V_2.$$

• If material has been removed, subtract the removed volume instead.
• For surface area, list only the surfaces visible from outside.
• Exclude surfaces where the component solids are joined internally.
• State the answer in square units for surface area or cubic units for volume.

Solving a melted-and-recast problem

• Identify the original solid and calculate its volume.
• Use the condition that no material is lost:

$$V_{\text{original}}=V_{\text{recast}}.$$

• Substitute the relevant formula for the new shape.
• Solve for the required dimension.
• Check that the final unit is appropriate.

Converting volume into capacity

• Express the volume in cubic centimetres where necessary.
• Use:

$$1\text{ litre}=1000{\text{ cm}}^3,1\text{ millilitre}=1{\text{ cm}}^3.$$

• Convert to litres or millilitres as required.
• Distinguish capacity from surface area: capacity concerns volume, not exposed area.

Where It Goes Wrong

• Using total surface area when only curved surface area or lateral surface area is required.
• Counting surfaces that are joined internally when finding the surface area of a composite solid.
• Confusing radius and diameter; the required relationship is $d=2r$.
• Using the cone’s perpendicular height in the curved-surface-area formula instead of the slant height $l$.
• Mixing units, such as using centimetres with metres, or failing to convert square and cubic units correctly.
• Giving a numerical answer without the required square or cubic units, or failing to check the formula dimensionally.

What Gets Asked

This material supports questions that require students to:

• Define surface area, curved surface area, total surface area, volume, radius, diameter, slant height, and composite solid.
• Calculate the lateral surface area, total surface area, or volume of a cuboid and cube.
• Calculate the curved surface area, total surface area, or volume of a cylinder.
• Calculate the curved surface area, total surface area, or volume of a cone.
• Use $l^2=r^2+h^2$ to find a cone’s slant height.
• Calculate the surface area or volume of a sphere and hemisphere.
• Convert between radius and diameter.
• Find the volume or exposed surface area of a composite solid.
• Solve problems involving a cylinder joined to a hemisphere.
• Apply conservation of volume when a solid is melted and recast into another shape, provided no material is lost.
• Convert cubic centimetres into litres or millilitres.
• Convert between metres and centimetres, square metres and square centimetres, or cubic metres and cubic centimetres.
• Distinguish practical uses of surface area and volume in painting, wrapping, polishing, filling, or melting objects.