Review of Length
Theory
This Year 10 maths page revises length and perimeter from Years 7 to 9: converting units of length, area and volume, prefixes such as mega and micro with answers in scientific notation, perimeters with missing sides, circles and semicircles, and the areas of borders and composite shapes.
The metric system builds every unit from a base unit (metre, gram, litre, second) and a prefix. The prefix is a power of ten: kilo means \(1000\), centi means one hundredth and milli means one thousandth.
To convert to a smaller unit, multiply: there will be more of them. To convert to a bigger unit, divide: there will be fewer.
Area units are squares, so the length factor is squared. Volume units are cubes, so it is cubed. Since \(1\text{ m} = 100\text{ cm}\), \(1\text{ m}^2 = 100^{2}\text{ cm}^2\) and \(1\text{ m}^3 = 100^{3}\text{ cm}^3\).
A hectare (ha) is the area of a square \(100\) m by \(100\) m, so \(1\text{ ha} = 10\,000\text{ m}^2\). Capacity measures how much a container holds: \(1\text{ mL} = 1\text{ cm}^3\) and \(1\text{ m}^3 = 1000\) L.
Very large and very small measurements, such as a wavelength in nanometres or power in megawatts, are often written in scientific notation, for example \(3.6 \times 10^{-7}\).
The perimeter is the distance around the outside of a shape. In a shape with right-angled corners, a missing side comes from the sides opposite it. The perimeter of a circle is its circumference. An arc is part of the circumference: a semicircle has half, a quadrant a quarter.
The area of a composite shape is found by adding the areas of simple pieces, or by taking a piece away from a bigger shape.
Metric prefixes:
| Prefix | Symbol | Means | Example |
|---|---|---|---|
| giga | G | \(10^{9}\) | \(1\text{ GB} = 10^{9}\) bytes |
| mega | M | \(10^{6}\) | \(1\text{ MW} = 10^{6}\) W |
| kilo | k | \(10^{3}\) | \(1\text{ km} = 1000\) m |
| centi | c | \(10^{-2}\) | \(1\text{ cm} = 0.01\) m |
| milli | m | \(10^{-3}\) | \(1\text{ mL} = 0.001\) L |
| micro | µ | \(10^{-6}\) | \(1\ \mu\text{g} = 10^{-6}\) g |
| nano | n | \(10^{-9}\) | \(1\text{ ns} = 10^{-9}\) s |
Conversion facts to know:
| Measure | Key facts |
|---|---|
| Length | \(1\text{ km} = 1000\text{ m}\), \(1\text{ m} = 100\text{ cm}\), \(1\text{ cm} = 10\text{ mm}\) |
| Area | \(1\text{ cm}^2 = 100\text{ mm}^2\), \(1\text{ m}^2 = 10\,000\text{ cm}^2\), \(1\text{ ha} = 10\,000\text{ m}^2\), \(1\text{ km}^2 = 100\text{ ha}\) |
| Volume | \(1\text{ cm}^3 = 1000\text{ mm}^3\), \(1\text{ m}^3 = 10^{6}\text{ cm}^3\) |
| Capacity | \(1\text{ L} = 1000\text{ mL}\), \(1\text{ mL} = 1\text{ cm}^3\), \(1\text{ m}^3 = 1000\text{ L}\) |
| Mass | \(1\text{ t} = 1000\text{ kg}\), \(1\text{ kg} = 1000\text{ g}\), \(1\text{ g} = 1000\text{ mg}\), \(1\text{ mg} = 1000\ \mu\text{g}\) |
| Time | \(1\text{ min} = 60\text{ s}\), \(1\text{ h} = 60\text{ min}\), \(1\text{ day} = 24\text{ h}\), \(1\text{ week} = 7\text{ days}\) |
Perimeter and circles:
How to convert a measurement
- Decide whether the new unit is bigger or smaller. Smaller unit: multiply. Bigger unit: divide.
- Find the factor. Step through one unit at a time (km to m to cm), and square or cube the length factor for area or volume.
- Calculate, and write very large or very small answers in scientific notation if asked.
- Check the size makes sense: more of a small unit, fewer of a big unit.
Perimeter and area problems
- Same units first. Before you add, multiply or divide two measurements, write them in the same unit.
- Perimeter of a composite shape: fill in every missing side, then walk round the outside once. An edge shared by two pieces is inside the shape and does not count.
- Curved edges: use the arc length. A semicircle on its own has a perimeter of the arc plus the diameter; a semicircle stuck on a side of a rectangle replaces that side.
- Area of a composite shape: split it into rectangles and add, or find a big rectangle and subtract the piece that is cut out.
- A path or border of width \(w\) all the way round: outer rectangle minus inner rectangle, where each outer measurement is \(2w\) longer.
- Cost: area in square metres times the price per square metre.
(i) \(0.86\) km to cm
(ii) \(4500\) mm to m
Move one unit at a time. Multiply going to a smaller unit, divide going to a bigger unit.
(i) km to m is \(\times 1000\), then m to cm is \(\times 100\).
| \(0.86 \times 1000\) | \(=\) | \(860\text{ m}\) |
| \(860 \times 100\) | \(=\) | \(\mathbf{86\,000}\text{ cm}\) |
(ii) mm to m is \(\div 1000\) (there are \(1000\) mm in a metre).
| \(4500 \div 1000\) | \(=\) | \(\mathbf{4.5}\text{ m}\) |
Sense check: centimetres are small, so there should be lots of them.
(i) \(2.5\text{ m}^2\) to \(\text{cm}^2\)
(ii) \(3400\text{ mm}^2\) to \(\text{cm}^2\)
(iii) \(0.75\text{ m}^3\) to \(\text{cm}^3\)
Square the length factor for area and cube it for volume.
(i) \(1\text{ m}^2 = 100 \times 100 = 10\,000\text{ cm}^2\)
| \(2.5 \times 10\,000\) | \(=\) | \(\mathbf{25\,000}\text{ cm}^2\) |
(ii) \(1\text{ cm}^2 = 10 \times 10 = 100\text{ mm}^2\), and mm² to cm² is a bigger unit, so divide.
| \(3400 \div 100\) | \(=\) | \(\mathbf{34}\text{ cm}^2\) |
(iii) \(1\text{ m}^3 = 100 \times 100 \times 100 = 10^{6}\text{ cm}^3\)
| \(0.75 \times 10^{6}\) | \(=\) | \(750\,000\) |
| \(\) | \(=\) | \(\mathbf{7.5 \times 10^{5}}\text{ cm}^3\) |
(i) Find the perimeter of the window, correct to two decimal places.
(ii) A circular window has the same perimeter. Find its radius, correct to two decimal places.
The top edge of the rectangle is inside the window, so it is not part of the perimeter. Use the arc instead: half the circumference of a circle with diameter \(1.2\) m.
(i)
| \(\text{arc}\) | \(=\) | \(\tfrac{1}{2} \times \pi \times 1.2\) |
| \(\) | \(=\) | \(1.8849\ldots\text{ m}\) |
| \(P\) | \(=\) | \(1.2 + 0.9 + 0.9 + 1.8849\ldots\) |
| \(\) | \(=\) | \(4.8849\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{4.88}\text{ m}\) |
(ii) Work backwards from \(C = 2\pi r\), using the unrounded perimeter.
| \(2\pi r\) | \(=\) | \(4.8849\ldots\) |
| \(r\) | \(=\) | \(\dfrac{4.8849\ldots}{2\pi}\) |
| \(\) | \(=\) | \(0.7774\ldots\) |
| \(\) | \(\approx\) | \(\mathbf{0.78}\text{ m}\) |
Only the three outside sides of the rectangle count: two heights and the bottom.
(i) Find the area of the deck.
(ii) Find the cost of the deck.
The deck is the big outer rectangle take away the sandpit. A border \(1.2\) m wide on both sides adds \(2 \times 1.2 = 2.4\) m to each measurement.
(i)
| \(\text{outer}\) | \(=\) | \((5 + 2.4) \times (3.2 + 2.4)\) |
| \(\) | \(=\) | \(7.4 \times 5.6 = 41.44\text{ m}^2\) |
| \(\text{sandpit}\) | \(=\) | \(5 \times 3.2 = 16\text{ m}^2\) |
| \(\text{deck}\) | \(=\) | \(41.44 - 16\) |
| \(\) | \(=\) | \(\mathbf{25.44}\text{ m}^2\) |
(ii)
| \(\text{cost}\) | \(=\) | \(25.44 \times \$85\) |
| \(\) | \(=\) | \(\mathbf{\$2162.40}\) |
Adding only \(1.2\) m to each side length gives too small a deck: the border is on both ends.
Common pitfalls
Frequently asked questions
How do you convert units in maths?
Work out whether the new unit is bigger or smaller. To change to a smaller unit multiply by the conversion factor; to change to a bigger unit divide. Step through one unit at a time if needed.
How many square centimetres are in a square metre?
There are \(10\,000\text{ cm}^2\) in \(1\text{ m}^2\), because a square metre is \(100\) cm by \(100\) cm and \(100 \times 100 = 10\,000\).
How many litres are in a cubic metre?
There are \(1000\) litres in \(1\text{ m}^3\). A litre is \(1000\text{ cm}^3\) and a cubic metre is \(10^{6}\text{ cm}^3\).
What is a hectare?
A hectare is \(10\,000\text{ m}^2\), the area of a square \(100\) m by \(100\) m. It is used for farms, parks and building blocks. To change square metres to hectares, divide by \(10\,000\).
What do micro and nano mean?
Micro (µ) means one millionth, \(10^{-6}\). Nano (n) means one billionth, \(10^{-9}\). So \(1\ \mu\text{m} = 10^{-6}\) m and \(1\text{ nm} = 10^{-9}\) m.
Do you multiply or divide to convert mm to m?
Divide. A metre is a bigger unit, so you get fewer of them. There are \(1000\) mm in a metre, so divide by \(1000\).
How do you find the perimeter of a semicircle?
Find half the circumference, \(\tfrac{1}{2}\pi d\), then add the diameter, because the straight edge is part of the outside of the shape.
How do you find the area of a path around a rectangle?
Find the area of the outer rectangle (each side is two path widths longer), then subtract the area of the inner rectangle.