Mensuration

The One Thing

Mensuration concerns the measurement of lengths, areas, surface areas, and volumes of three-dimensional solids. Reliable solutions depend on identifying the solid and the exact quantity required—curved surface area, total surface area, or volume—before selecting a formula and applying consistent units.

Definitions and Results

• Mensuration: The branch of mathematics concerned with measuring lengths, areas, surface areas, and volumes.
• Radius: The distance from the centre of a circular base or sphere to its boundary, usually denoted by $r$.
• Diameter: The distance across a circle through its centre; $d=2r$.
• Height: The perpendicular distance between the bases of a solid, usually denoted by $h$.
• Slant height: The inclined length from the vertex to the edge of the circular base of a cone, usually denoted by $l$.
• Curved Surface Area: The area of only the curved surface of a solid, excluding its circular or other plane bases.
• Total Surface Area: The sum of the curved surface area and the areas of all plane surfaces.
• Volume: The amount of space occupied by a solid, expressed in cubic units.
• Cylinder: A solid with two equal, parallel circular bases joined by a curved surface.
• Cone: A solid with one circular base and a vertex connected to every point on the boundary of the base.
• Sphere: A perfectly round solid whose every point on its surface is at the same distance from its centre.
• Hemisphere: Half of a sphere, obtained by cutting a sphere through its centre.
• Frustum of a cone: The portion of a cone left after its top is cut off by a plane parallel to the base.
• Capacity: The volume a container can hold, commonly measured in litres or millilitres.
• Cylinder formulae: For radius $r$ and height $h$,

$$\text{curved surface area}=2\pi rh,$$ $$\text{total surface area}=2\pi r(h+r),$$ $$\text{volume}=\pi r^{2}h.$$

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

$$l^2=r^2+h^2,$$ $$\text{curved surface area}=\pi rl,$$ $$\text{total surface area}=\pi r(l+r),$$ $$\text{volume}=\frac{1}{3}\pi r^{2}h.$$

• Sphere formulae: For radius $r$,

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

• Hemisphere formulae: For radius $r$,

$$\text{curved surface area}=2\pi r^2,$$ $$\text{total surface area}=3\pi r^2,$$ where the total surface area includes the circular base, and $$\text{volume}=\frac{2}{3}\pi r^3.$$

• Frustum formulae: For larger radius $R$, smaller radius $r$, height $h$, and slant height $l$,

$$l^2=h^2+(R-r{)}^2,$$ $$\text{curved surface area}=\pi (R+r)l,$$ $$\text{total surface area}=\pi (R+r)l+\pi R^2+\pi r^2,$$ $$\text{volume}=\frac{1}{3}\pi h(R^2+r^2+Rr).$$

• Units: 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 ${\text{m}}^3$.
• Value of $\pi$: Use $\pi =\frac{22}{7}$ or $3.14$, according to the question, and retain the required accuracy in the final answer.
• Capacity conversions:

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

• Melted and recast solids: When a solid is melted and recast without loss, its volume remains unchanged.
• Combined solids: Add or subtract the volumes and surface areas of the relevant parts, taking care not to count joined internal surfaces.
• Hollow solids: The volume of material is found by subtracting the volume of the inner part from the volume of the outer part.
• Cone–cylinder relationship: The volume of a cone is one-third the volume of a cylinder having the same base radius and height.
• Pythagorean relationship: The Pythagorean theorem connects the height, radius, and slant height of a right circular cone or frustum.
• Dimensional dependence: Surface area depends on squared measurements, whereas volume depends on cubed measurements; unit conversions must therefore be handled carefully.

Worked Methods

Selecting the required measurement

• Identify the solid: cylinder, cone, sphere, hemisphere, frustum, or a combination of solids.
• Determine whether the question requires curved surface area, total surface area, or volume.
• Identify the relevant dimensions, including radius, diameter, height, slant height, larger radius, or smaller radius.
• Convert all measurements to consistent units.
• Select and substitute into the appropriate formula.
• State the answer with square units for area or cubic units for volume.
• Check that the result has practical meaning.

For a cylinder, use $2\pi rh$ for curved surface area, $2\pi r(h+r)$ for total surface area, and $\pi r^{2}h$ for volume. The circular bases are excluded from curved surface area but included in total surface area.

Finding a cone’s slant height

For a cone with radius $r$ and perpendicular height $h$:

• Use the right-triangle relationship

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

• Substitute the given radius and height.
• Take the positive square root to obtain $l$.
• Use $l$ in

$$\text{curved surface area}=\pi rl$$ or $$\text{total surface area}=\pi r(l+r).$$

• Use the perpendicular height $h$, not the slant height, in

$$\text{volume}=\frac{1}{3}\pi r^{2}h.$$

Finding a frustum’s slant height

For a frustum with larger radius $R$, smaller radius $r$, and height $h$:

• Find the difference between the radii, $R-r$.
• Apply

$$l^2=h^2+(R-r{)}^2.$$

• Take the positive square root to find $l$.
• Use $l$ in

$$\text{curved surface area}=\pi (R+r)l.$$

• Add the areas of both circular ends for total surface area:

$$\text{total surface area}=\pi (R+r)l+\pi R^2+\pi r^2.$$

• Use the height $h$ in the volume formula:

$$\text{volume}=\frac{1}{3}\pi h(R^2+r^2+Rr).$$

Solving composite-solid problems

• Divide the figure into familiar solids, such as cylinders, cones, spheres, hemispheres, or frustums.
• Identify which surfaces are exposed and which surfaces are joined internally.
• Calculate the required volume or surface area of each relevant part.
• Add volumes when parts are joined externally.
• Subtract volumes when a portion is removed or when finding the material in a hollow solid.
• Do not count joined internal surfaces in the external surface area.
• Give the result in the appropriate square or cubic units.

Solving melted-and-recast problems

• Identify the original solid and the recast solid.
• Calculate the volume of the original solid.
• Set the volume of the original solid equal to the volume of the recast solid:

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

• Substitute the relevant formula for each solid.
• Solve for the unknown dimension.
• This equality applies because the solid is melted and recast without loss.

Solving capacity problems

• Calculate the volume of the container using the appropriate solid formula.
• Express the volume in a convenient unit.
• Convert using

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

• State the capacity in litres or millilitres as required.

Where It Goes Wrong

• Using curved surface area when the question requires total surface area, or including circular bases when only the curved portion is required.
• Using the diameter in a formula that requires the radius without first applying $d=2r$.
• Using the slant height in the cone’s volume formula; the volume requires the perpendicular height $h$, whereas surface-area formulae use the slant height $l$.
• Forgetting that the total surface area of a hemisphere, $3\pi r^2$, includes its circular base, while its curved surface area, $2\pi r^2$, does not.
• Failing to use $R-r$ in the frustum relationship

$$l^2=h^2+(R-r{)}^2.$$

• Mixing units or reporting area in cubic units and volume in square units; surface area uses squared units and volume uses cubed units.

What Gets Asked

This material supports questions that require students to:

• Define mensuration terms such as radius, diameter, height, slant height, curved surface area, total surface area, volume, capacity, cylinder, cone, sphere, hemisphere, and frustum of a cone.
• Calculate the curved surface area, total surface area, or volume of a cylinder, cone, sphere, hemisphere, or frustum.
• Find a cone’s slant height using $l^2=r^2+h^2$.
• Find a frustum’s slant height using $l^2=h^2+(R-r{)}^2$.
• Distinguish between curved surface area and total surface area.
• Solve problems involving combined solids, including the treatment of joined internal surfaces.
• Find the volume of material in a hollow solid by subtracting the inner volume from the outer volume.
• Solve melted-and-recast problems using conservation of volume.
• Convert between cubic units and capacity units.
• Apply $1\text{ litre}=1000{\text{ cm}}^3$, $1\text{ millilitre}=1{\text{ cm}}^3$, and $1{\text{ m}}^3=1000\text{ litres}$.
• Explain why the volume of a cone is one-third the volume of a cylinder with the same base radius and height.
• Check whether an answer has the correct dimensions and practical interpretation.