Surface Areas and Volumes

The One Thing

Problems involving combinations of solids are solved by separating the object into familiar three-dimensional shapes. Add or subtract component volumes as appropriate, but calculate surface area only from the exposed external surfaces, excluding all common or joining surfaces.

Definitions and Results

• Combination of solids: A three-dimensional object made by joining two or more standard solids, such as a cylinder with a hemisphere or a cone with a cylinder.

• Surface area: The total area of all exposed surfaces of a solid, measured in square units.

• Volume: The amount of space occupied by a solid, measured in cubic units.

• Exposed surface: A part of the outer surface that can be seen or touched after solids are joined.

• Common or joining surface: The surface through which two solids are attached. It is internal and is not included in the external surface area.

• Curved surface area: The area of the curved part of a solid, excluding its circular or flat bases.

• Total surface area: The sum of all exposed curved and plane surfaces of a solid.

• Radius and diameter: The radius is the distance from the centre to the boundary of a circle. The diameter is twice the radius:

$$d=2r.$$

• Slant height: The distance measured along the curved surface of a cone from its vertex to the edge of its circular base:

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

• Unit conversion: All lengths, areas, and volumes must use compatible units:

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

• Use of $\pi$: Use $\pi =\frac{22}{7}$ or $\pi =3.14$, according to the instruction in the question.

• Cuboid:

$$V=lbh,$$ $$\text{TSA}=2(lb+bh+hl),$$ where $l$, $b$, and $h$ are the length, breadth, and height.

• Cube:

$$V=a^3,$$ $$\text{TSA}=6a^2,$$ where $a$ is the edge length.

• Cylinder:

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

• Cone:

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

• Sphere:

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

• Hemisphere:

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

• Cylinder joined to a hemisphere along a circular face: The joining circular face is excluded. The exposed surface area may therefore be

$$2\pi rh+2\pi r^2.$$

• Cone joined to a cylinder along a circular base: The exposed surface area is the sum of the cone’s curved surface area, the cylinder’s curved surface area, and any exposed base.

• Solid formed by removal: If one shape is removed from another,

$$V_{\text{remaining}}=V_{\text{original}}-V_{\text{removed part}}.$$

• Hollow solid: The volume of material is

$$V_{\text{material}}=V_{\text{outer solid}}-V_{\text{inner cavity}}.$$

• Joined solids and volume: When solids are attached, their joining surface has no volume, so the total volume is usually the sum of their component volumes.

• Units: Surface area must be expressed in square units, whereas volume must be expressed in cubic units.

Worked Methods

Method 1: Finding the surface area of a combined solid

• Examine or draw the figure.
• Identify the component solids, such as cuboids, cylinders, cones, spheres, or hemispheres.
• Label all dimensions, including radii, diameters, heights, and slant heights.
• Determine which surfaces remain visible after the solids are joined.
• Exclude every common or joining surface.
• Select the appropriate curved-surface-area and plane-surface-area formulae.
• Add only the exposed areas.
• Express the result in square units.

Example: Cylinder joined to a hemisphere

For a cylinder joined to a hemisphere along a circular face:

• Identify the cylinder’s curved surface:

$$2\pi rh.$$

• Identify the hemisphere’s exposed curved surface:

$$2\pi r^2.$$

• Exclude the circular face joining the cylinder and hemisphere.
• Add the exposed areas:

$$\text{Exposed surface area}=2\pi rh+2\pi r^2.$$

The joining circular face must not be counted because it is internal.

Example: Cone joined to a cylinder

For a cone joined to a cylinder along a circular base:

• Calculate the cone’s curved surface area:

$$\pi rl.$$

• Calculate the cylinder’s curved surface area:

$$2\pi rh.$$

• Identify any circular base that remains exposed.
• Add the cone’s curved surface area, the cylinder’s curved surface area, and the exposed base area.
• Exclude the circular base through which the cone and cylinder are joined.
• Express the answer in square units.

Method 2: Finding the volume of a combined solid

• Examine the figure and identify its component solids.
• Convert all dimensions into the same unit.
• Select the volume formula for each component.
• Add the component volumes when the solids are attached:

$$V_{\text{combined}}=V_1+V_2+\cdots .$$

• Express the answer in cubic units.

For example, if a combined object consists of a cylinder and a hemisphere, calculate $$V_{\text{cylinder}}=\pi r^{2}h$$ and $$V_{\text{hemisphere}}=\frac{2}{3}\pi r^3,$$ then add them: $$V_{\text{combined}}=\pi r^{2}h+\frac{2}{3}\pi r^3.$$

The joining surface does not need to be subtracted because it has no volume.

Method 3: Finding the volume of a solid with a cavity or cut-out

• Identify the original outer solid.
• Identify the shape removed from it, such as a cavity or cut-out.
• Calculate the volume of the original solid.
• Calculate the volume of the removed part.
• Subtract:

$$V_{\text{remaining}}=V_{\text{original}}-V_{\text{removed part}}.$$

• If the object is hollow, interpret the result as the volume of material:

$$V_{\text{material}}=V_{\text{outer solid}}-V_{\text{inner cavity}}.$$

• Express the answer in cubic units.

Method 4: Converting dimensions and checking units

• Inspect the units of every given dimension.
• Convert all lengths to one unit before using a formula. For example:

$$1\text{ m}=100\text{ cm}.$$

• Apply the corresponding area or volume conversion when necessary:

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

• Use the selected formula.
• Check that a surface-area answer is in square units and a volume answer is in cubic units.

Method 5: General diagram-based strategy

• Draw or examine the diagram carefully.
• Split the object into familiar solids.
• Label every relevant dimension.
• Determine whether the problem requires surface area, volume, or both.
• For surface area, list only the visible or exposed regions.
• For volume, decide whether to add component volumes or subtract a removed region.
• Select and apply the relevant formulae.
• Check that each required region has been included exactly once.
• Simplify the result using the instructed value of $\pi$.
• Verify the final units.

Where It Goes Wrong

• Adding the complete total surface areas of separate solids without removing the common or joining surfaces; this can count internal surfaces twice.

• Including a circular face where two solids are attached, even though that face is not exposed.

• Adding volumes in a hollow-solid or cut-out problem instead of subtracting the volume of the inner cavity or removed part:

$$V_{\text{remaining}}=V_{\text{original}}-V_{\text{removed part}}.$$

• Using inconsistent units, such as combining metres and centimetres before applying a formula.

• Confusing radius and diameter; the required relationship is

$$d=2r.$$

• Giving surface area in cubic units or volume in square units instead of using square units for area and cubic units for volume.

What Gets Asked

This material supports questions requiring students to:

• Calculate the exposed surface area of a cylinder joined to a hemisphere, excluding the circular joining face.
• Calculate the exposed surface area of a cone joined to a cylinder, including only the relevant curved surfaces and exposed bases.
• Find the volume of a combined solid by adding the volumes of its component solids.
• Find the volume remaining after a shape is removed from another shape.
• Calculate the material volume of a hollow solid by subtracting the inner cavity from the outer solid.
• Use the formulae for cuboids, cubes, cylinders, cones, spheres, and hemispheres.
• Determine a cone’s slant height using

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

• Convert lengths, areas, and volumes into compatible units before calculation.
• Identify exposed and hidden surfaces from a diagram.
• State final answers with the correct square or cubic units and with $\pi =\frac{22}{7}$ or $\pi =3.14$, as instructed.