Measurement

The One Thing

Measurement problems are solved by identifying the required quantity, selecting an appropriate formula, converting all measurements into compatible units, and checking the result through units, estimation and scale reasoning. Surface area describes the outside covering of a three-dimensional object, whereas volume describes the space it occupies.

Definitions and Results

• Length: The distance between two points, measured in units such as millimetres, centimetres, metres or kilometres.
• Area: The amount of two-dimensional surface inside a boundary, measured in square units such as cm² or m².
• Volume: The amount of three-dimensional space occupied by an object, measured in cubic units such as cm³ or m³.
• Surface area: The total area of every outside surface of a three-dimensional object.
• Capacity: The amount a container can hold, commonly measured in litres or millilitres.
• Prism: A three-dimensional object with uniform cross-sections along its length. Its volume is the area of the cross-section multiplied by the length.
• Cylinder: A prism with circular bases. Its volume and surface area depend on its radius and height.
• Composite object: A shape formed by combining or removing simpler two-dimensional or three-dimensional shapes.
• Scale factor: The multiplier used to enlarge or reduce a measurement. Lengths scale by $k$, areas by $k^2$, and volumes by $k^3$.
• Unit conversion: Changing a measurement into an equivalent unit, such as converting metres to centimetres or cubic metres to litres.
• Dimensional reasoning: Checking the type of unit produced by a calculation. Perimeter and circumference use linear units, area and surface area use square units, and volume uses cubic units.
• Accuracy and precision: Accuracy is closeness to the true value; precision is the level of detail or consistency in a measurement.
• Length conversions:

$1\text{ m}=100\text{ cm}$, $1\text{ cm}=10\text{ mm}$, and $1\text{ km}=1000\text{ m}$.

• Area conversions:

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

• Volume conversions:

$1{\text{ m}}^3=1,000,000{\text{ cm}}^3$ and $1{\text{ cm}}^3=1000{\text{ mm}}^3$.

• Capacity conversions:

$1\text{ L}=1000\text{ mL}$, $1\text{ L}=1{\text{ dm}}^3$, and $1\text{ mL}=1{\text{ cm}}^3$.

• Rectangle area:

$$A=lw$$

• Triangle area:

$$A=\frac{1}{2}bh$$

• Parallelogram area:

$$A=bh$$

• Trapezium area:

$$A=\frac{1}{2}(a+b)h$$ where $a$ and $b$ are the parallel sides.

• Circle circumference:

$$C=2\pi r\text{or}C=\pi d$$

• Circle area:

$$A=\pi r^2$$

• Rectangular prism volume:

$$V=lwh$$

• General prism volume:

$$V=\text{area of cross-section}\times \text{length}$$

• Cylinder volume:

$$V=\pi r^{2}h$$

• Rectangular prism surface area:

$$SA=2lw+2lh+2wh$$

• Cylinder surface area, including both circular ends:

$$SA=2\pi r^2+2\pi rh$$

• Use of radius: Circle formulas require the radius. If the diameter is given, use

$$r=\frac{d}{2}.$$

• Use of $\pi$: Use $\pi$ approximately as $3.14159$, or use the $\pi$ key on a calculator, unless another value is specified.
• Rounding: Round only at the end of a calculation unless an estimate is specifically requested.
• Reasonableness: Check answers using estimation, unit analysis and comparison with the original dimensions.

Worked Methods

1. Converting length units

• Identify the starting and required units.
• Use the relevant conversion relationship:

- $1\text{ m}=100\text{ cm}$ - $1\text{ cm}=10\text{ mm}$ - $1\text{ km}=1000\text{ m}$

• Multiply or divide by the conversion factor as appropriate.
• Label the result with the new unit.

2. Converting area units

• Identify the linear conversion factor.
• Square the conversion factor because area is two-dimensional.
• Apply the squared relationship:

- $1{\text{ m}}^2=10,000{\text{ cm}}^2$ - $1{\text{ cm}}^2=100{\text{ mm}}^2$

• Check that the final unit is a square unit.

3. Converting volume and capacity units

• Identify the linear conversion factor.
• Cube the conversion factor because volume is three-dimensional.
• Apply the appropriate relationship:

- $1{\text{ m}}^3=1,000,000{\text{ cm}}^3$ - $1{\text{ cm}}^3=1000{\text{ mm}}^3$

• For capacity, use:

- $1\text{ L}=1000\text{ mL}$ - $1\text{ L}=1{\text{ dm}}^3$ - $1\text{ mL}=1{\text{ cm}}^3$

4. Calculating two-dimensional area

• Identify the shape.
• Select the correct formula:

- Rectangle: $A=lw$ - Triangle: $A=\frac{1}{2}bh$ - Parallelogram: $A=bh$ - Trapezium: $A=\frac{1}{2}(a+b)h$ - Circle: $A=\pi r^2$

• Convert measurements to compatible units.
• Substitute the measurements into the formula.
• Calculate and give the answer in square units.

5. Calculating circumference

• Determine whether the radius or diameter is provided.
• If the diameter is provided, calculate $r=d/2$ if using $C=2\pi r$.
• Use either

$$C=2\pi r$$ or $$C=\pi d.$$

• Give the answer in linear units.

6. Calculating the volume of a rectangular prism

• Identify the length, width and height.
• Convert them to compatible units.
• Use

$$V=lwh.$$

• Substitute the measurements.
• Give the answer in cubic units.

7. Calculating the volume of a general prism

• Identify the area of the uniform cross-section.
• Identify the length of the prism.
• Use

$$V=\text{area of cross-section}\times \text{length}.$$

• Calculate the cross-sectional area if it is not already given.
• Multiply by the length and state the result in cubic units.

8. Calculating the volume of a cylinder

• Identify the radius and height.
• If the diameter is given, use $r=d/2$.
• Convert the radius and height to compatible units.
• Use

$$V=\pi r^{2}h.$$

• Substitute the values and calculate using $\pi$ approximately as $3.14159$ or the calculator’s $\pi$ key.
• Round only at the end and state the answer in cubic units.

9. Calculating surface area

For a rectangular prism:

• Identify $l$, $w$ and $h$.
• Use

$$SA=2lw+2lh+2wh.$$

• Add the areas of all six rectangular faces.
• State the result in square units.

For a cylinder:

• Identify the radius and height.
• Calculate the two circular ends and the curved surface.
• Use

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

• State the result in square units.

10. Solving composite-object problems

• Divide the composite object into familiar solids.
• Calculate the area or volume of each part.
• Combine the results by addition.
• If a section has been removed, subtract its area or volume.
• Check that the final unit is square units for surface area or cubic units for volume.

11. Applying scale factors

• Identify the scale factor $k$.
• Multiply each length by $k$.
• Multiply areas by $k^2$.
• Multiply volumes by $k^3$.
• Ensure that the scaling rule matches the quantity being measured.

12. Checking a result

• Check the dimensions and units used in the formula.
• Confirm that the final unit matches the quantity:

- linear units for perimeter and circumference; - square units for area and surface area; - cubic units for volume.

• Estimate the expected size of the answer.
• Compare the result with the original dimensions.
• Consider whether the measurements are approximate and whether the rounding is suitable.

Where It Goes Wrong

• Using a diameter in a formula that requires a radius; the required conversion is $r=d/2$.
• Converting area or volume using the original length factor instead of squaring or cubing it.
• Substituting measurements in incompatible units before converting them.
• Treating surface area and volume as interchangeable, even though one measures outside covering and the other measures internal space.
• Omitting a face of a rectangular prism, a circular end of a cylinder, or a removed section in a composite object.
• Rounding intermediate values rather than rounding only at the end, unless an estimate is specifically requested.

What Gets Asked

This material supports questions requiring students to:

• Convert lengths, areas, volumes and capacities between metric units.
• Select and apply formulas for rectangles, triangles, parallelograms, trapezia and circles.
• Calculate circumference using $C=2\pi r$ or $C=\pi d$.
• Calculate the volume of rectangular prisms, general prisms and cylinders.
• Calculate the surface area of rectangular prisms and cylinders.
• Find the area or volume of composite objects by combining or subtracting familiar shapes.
• Use the radius when circle measurements are given as diameters.
• Apply scale factors to lengths, areas and volumes.
• Identify errors through dimensional reasoning, estimation and comparison with the original dimensions.
• Present answers with appropriate units, rounding and sufficient working.