Applications of Measurement

Metric unit conversions, scientific notation with prefixes, perimeter and area including Pythagoras' theorem and irregular blocks, surface area, volume and capacity of prisms, cylinders, cones, pyramids and spheres, and the trapezoidal rule, with worked examples.

1What you need to know
2Units and conversions
3Perimeter, area and Pythagoras
4Surface area, volume and capacity
5The trapezoidal rule
6Common mistakes
7Exam tips
8Links to other topics

1.What you need to know

Applications of measurement is one of the most heavily weighted Year 11 focus areas in Standard 2 (4 marks in the sample HSC paper, and often more in past papers). Most questions are practical: a garden, a tank, a block of land.

Units: convert metric units of length, area, volume, capacity and mass, and between volume and capacity.
Scientific notation with prefixes from nano to tera.
Perimeter and area of triangles, rectangles, parallelograms, trapeziums, circles, sectors and composite shapes.
Pythagoras' theorem in right-angled triangles.
Irregular blocks of land by dissection into triangles, rectangles and trapeziums.
Surface area of prisms, cylinders, spheres and composite solids.
Volume and capacity of prisms, cylinders, spheres, cones, pyramids and composite solids.
The trapezoidal rule, used up to four times, for area, volume and capacity.

Measurement errors (absolute error, percentage error, limits of accuracy) were in the old syllabus but are not in this focus area.

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2.Units and conversions

1 cm 1 m 1 km = 10 mm = 100 cm = 1000 m

For area, square the length conversion; for volume, cube it.

1 cm^{2} 1 m^{2} 1 ha 1 km^{2} = 100 mm^{2} = 10000 cm^{2} = 10000 m^{2} = 1000000 m^{2}

1 cm^{3} 1 m^{3} = 1000 mm^{3} = 1000000 cm^{3}

Capacity is how much a container holds. The links between volume and capacity are the most examined conversions:

1 cm^{3} 1 m^{3} 1000 kL = 1 mL = 1000 L = 1 kL = 1 ML

Mass: 1 g = 1000 mg, 1 kg = 1000 g, 1 t = 1000 kg. To go to a smaller unit, multiply; to go to a bigger unit, divide.

Prefixes and scientific notation

Prefix	Symbol	Power
tera	T	$1 0^{12}$
giga	G	$1 0^{9}$
mega	M	$1 0^{6}$
kilo	k	$1 0^{3}$
centi	c	$1 0^{- 2}$
milli	m	$1 0^{- 3}$
micro	µ	$1 0^{- 6}$
nano	n	$1 0^{- 9}$

So 4.7 gigabytes is $4.7 \times 1 0^{9}$ bytes, and 350 nanometres is $350 \times 1 0^{- 9} = 3.5 \times 1 0^{- 7}$ metres. In scientific notation the first number must be at least 1 and less than 10.

You may be asked to give the answer to a number of significant figures: 0.000 004 87 m to 2 significant figures is $4.9 \times 1 0^{- 6}$ m (4.9 micrometres).

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3.Perimeter, area and Pythagoras

triangle: A parallelogram: A trapezium: A circle: A C = \frac{1}{2} bh = bh = \frac{1}{2} h (a + b) = π r^{2} = 2 π r

A sector is a slice of a circle with angle $θ$ at the centre:

A l = \frac{θ}{360} \times π r^{2} = \frac{θ}{360} \times 2 π r

The perimeter of a sector is the arc length $l$ plus two radii. For a sector of radius 3 m and angle 110°, the arc is 5.76 m and the perimeter is $5.76 + 6 = 11.76$ m, while the area is 8.64 m² (2 d.p.).

Pythagoras' theorem: in a right-angled triangle with hypotenuse $c$ (the longest side, opposite the right angle),

c^{2} = a^{2} + b^{2}

Worked example: edging a triangular garden bed

A garden bed is a right-angled triangle with the two shorter sides 2.4 m and 3.2 m. Find the length of edging needed to go all the way around, and the area of the bed.

c^{2} c = 2. 4^{2} + 3. 2^{2} = 5.76 + 10.24 = 16 = 4 m

P A = 2.4 + 3.2 + 4 = 9.6 m = \frac{1}{2} \times 2.4 \times 3.2 = 3.84 m^{2}

The bed needs 9.6 m of edging and has an area of 3.84 m². In a right-angled triangle, the two shorter sides are the base and perpendicular height.

Irregular blocks of land

Split an irregular block into triangles, rectangles and trapeziums, find each area, then add. Mark the lengths and right angles you know on the diagram first, and check that the pieces cover the whole block without overlapping.

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4.Surface area, volume and capacity

Surface area is the total area of all faces. For a prism, add the area of every face (drawing the net helps). For curved solids:

closed cylinder: S A sphere: S A = 2 π r^{2} + 2 π r h = 4 π r^{2}

Read the question carefully: an open tank has no lid, so leave out one circle or one face.

Volume formulas:

prism: V cylinder: V pyramid: V cone: V sphere: V = A h = π r^{2} h = \frac{1}{3} A h = \frac{1}{3} π r^{2} h = \frac{4}{3} π r^{3}

Here $A$ is the area of the base and $h$ the perpendicular height. The HSC reference sheet lists the harder formulas, but you need to know which one to use and what each letter means.

Worked example: capacity of a silo

A grain silo is a cylinder of radius 2 m and height 6 m, with a cone on top of radius 2 m and height 1.5 m. Find its volume in cubic metres and its capacity in kilolitres, correct to one decimal place.

V_{cyl} V_{cone} = π \times 2^{2} \times 6 = 24 π = \frac{1}{3} \times π \times 2^{2} \times 1.5 = 2 π

V = 24 π + 2 π = 26 π = 81.68 \dots m^{3} \approx 81.7 m^{3}

Since 1 m³ holds 1 kL, the capacity is about 81.7 kL, or 81 681 L. Keep $π$ exact (or use the calculator's stored value) until the last step.

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5.The trapezoidal rule

The trapezoidal rule estimates an area with a curved edge by treating it as trapeziums:

A \approx \frac{h}{2} (d_{f} + d_{l})

Here $d_{f}$ is the first measurement, $d_{l}$ the last measurement and $h$ the distance between them. For more than one strip, apply the rule to each strip and add the results. The syllabus uses up to four applications.

Worked example: area and volume of a pond

A pond is measured across at 12 m intervals along a straight line. The widths are 5.6 m, 9.0 m and 7.2 m. Use two applications of the trapezoidal rule to estimate its surface area. If the average depth is 1.5 m, estimate the volume of water in kilolitres.

A \approx \frac{12}{2} (5.6 + 9.0) + \frac{12}{2} (9.0 + 7.2) = 87.6 + 97.2 = 184.8 m^{2}

V \approx 184.8 \times 1.5 = 277.2 m^{3}

The pond holds about 277.2 kL of water. Notice that the middle width (9.0 m) is used in both strips. The rule also works on a cross-section: estimate the area of the cross-section, then multiply by the length to find a volume.

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6.Common mistakes

Converting square or cubic units with the length factor: 1 m² is 10 000 cm², not 100 cm².
Using the diameter as the radius.
Using the slant height instead of the perpendicular height in a cone or pyramid volume.
Forgetting the $\frac{1}{3}$ for cones and pyramids, or cubing in the wrong place for a sphere.
Adding the hypotenuse squared instead of subtracting when finding a shorter side: $a^{2} = c^{2} - b^{2}$ .
Using the wrong $h$ in the trapezoidal rule. It is the gap between measurements, not the total length.
Leaving a capacity in m³ when the question asks for litres.

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7.Exam tips

Write the formula, then the substitution, then the answer with units. The substitution usually earns a mark even if the arithmetic slips.
For composite shapes, write a short plan in words ("cylinder + cone") before calculating.
Check rounding instructions: "correct to 1 decimal place" or "nearest square metre" are common.
Draw or add to the diagram: mark the radius, height and any right angles before you start.

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8.Links to other topics

Trigonometry (Year 12): right-angled and non-right-angled triangles extend Pythagoras' theorem, with areas found using sine.
Ratios and rates (Year 12): scale drawings, rates of flow and energy use build on these unit conversions.
Formulas and equations: every measurement formula is substitution.
Time and location: both use rounding and unit conversions carefully.

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Test yourself

Practise 11 exam-style questions on Applications of Measurement, each with instant marking and a worked solution.

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Applications of Measurement

Contents

1.What you need to know

2.Units and conversions

Prefixes and scientific notation

3.Perimeter, area and Pythagoras

Worked example: edging a triangular garden bed

Irregular blocks of land

4.Surface area, volume and capacity

Worked example: capacity of a silo

5.The trapezoidal rule

Worked example: area and volume of a pond

6.Common mistakes

7.Exam tips

8.Links to other topics

Test yourself