Today we compare. Which is longer? Which is shorter? Which is heavier? Which is lighter?
Pencils come in different lengths. Line them up to compare. Photo: MichaelMaggs, CC BY-SA 3.0, via Wikimedia Commons.
Longer and shorter
To compare two things, line them up at one end. Then look at the other end.
✏️✏️✏️✏️✏️ 🖍️🖍️🖍️
The pencil goes further. The pencil is longer. The crayon is shorter.
If they end at the same place, they are the same length.
Tall and short
People and trees can be tall or short. Stand back to back with someone on flat ground. Who is taller?
Heavy and light
Hold one thing in each hand. The one that pulls your hand down more is heavier. The other is lighter.
A watermelon is heavier than an apple. A feather is very light.
A watermelon is much heavier than an apple. Hold an apple: is it light or heavy? Photo: Peculiar grace, CC0, via Wikimedia Commons.
Big is not always heavy
A big balloon is light. A small stone can be heavy. We must feel or use a balance to check, not just look.
On a balance or a see-saw, the heavier side goes down.
A feather is very light. You can hardly feel it in your hand. Photo: Joao Estevao Andrade de Freitas, Public domain, via Wikimedia Commons.
Try it
TRY ITCompare it
Pick a game. Then tap your answer.
Let’s practise
Point, say or draw. Open each answer after trying.
1. Find a spoon and a fork. Line them up at one end. Which is longer?
Show answer
Answers will vary. Check that the ends were lined up first.
2. Stand back to back with an adult. Who is taller?
Show answer
Usually the adult. Say: the adult is taller, I am shorter.
3. Hold a shoe in one hand and a sock in the other. Which is heavier?
Show answer
Usually the shoe is heavier.
4. Which can be small but heavy? (a) a balloon (b) a stone (c) a feather
Show answer
(b) a stone.
Quick check
Try these three on your own first. An adult may read the words.
1. Two strips start at the same line. The red strip goes further. Which is longer?
The red strip
The other strip
They are the same
Show answer
The red strip. The one that reaches further from the same start is longer.
2. Which is heavier?
A feather
A watermelon
A leaf
Show answer
A watermelon. A watermelon is heavy. Feathers and leaves are light.
3. Before we compare two lengths, what do we do?
Line them up at one end
Pick the brighter one
Close our eyes
Show answer
Line them up at one end. Line up one end, then look at the other end.
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Maths lesson 14: Long and Short, Heavy and Light — worksheet
Point, draw or tell an adult. An adult can write for you.
Draw a long snake and a short snake. Circle the longer one.
Draw something heavy and something light. Write or say heavy and light.
Draw a see-saw with a heavy thing on one side. Which side goes down?
Say these words with an adult. Can you use each one in a sentence?
long · longer · short · shorter · tall · heavy · heavier · light · lighter · compare · balance
Try it at home
Line up three shoes from one end. Find the longest and the shortest.
Heft two things from the kitchen, like a tin and a packet of crackers. Guess first, then feel.
Make a simple balance with a coat hanger and two bags. Which bag goes down?
For parents and teachers
After this lesson, your child should be able to:
line up two objects from one end and say which is longer, shorter or the same length
hold two objects and say which is heavier or lighter
give one example where a smaller object is heavier than a bigger one
Use safe, real objects for hefting, not only pictures. Comparing directly comes before measuring with units such as centimetres or kilograms.
Accept pointing, speaking, signing or communication aids. An adult can scribe. Observe the hands-on work as well as the Quick check; the badge records supported completion only.
In Lesson 20 you chose instruments and units and read scales. In Form 1 you met metric and imperial units of length, grams and kilograms and perimeter and area. Now we put it all together and convert from one unit to another. You will need this every day: a recipe asks for 750 g of flour but the bag says 2 kg, a cousin in the United States says the beach is 12 miles away, a man at the hardware tells you the board is 8 feet long, or a tiler needs to know how many tiles cover a floor measured in square metres.
Learning outcomes
By the end of this lesson you should be able to:
2.5.1.5 – Change a measurement from one metric unit to another (length, mass and capacity).
2.5.1.6 – Change a measurement between metric and imperial units using the unitary method.
2.5.1.7 – Change an area from one unit to another, for example from m² to cm².
The metric system works in tens
Metric units are built on our base-ten number system. Each unit has a prefix that tells you how big it is compared with the base unit (metre, gram or litre):
Prefix
Meaning
Example
kilo-
1,000 times
1 km = 1,000 m
centi-
one hundredth
100 cm = 1 m
milli-
one thousandth
1,000 mm = 1 m
The key facts to know:
Length: 10 mm = 1 cm, 100 cm = 1 m, 1,000 m = 1 km.
Mass: 1,000 mg = 1 g, 1,000 g = 1 kg, 1,000 kg = 1 tonne (t).
Capacity: 1,000 ml = 1 l. Also, 1 ml takes up 1 cm³ of space, so a 1 litre carton holds 1,000 cm³.
Figure 1 – The length staircase. Going down to a smaller unit, multiply. Going up to a larger unit, divide.
There is one simple rule:
Changing to a smaller unit? You need more of them, so multiply.
Changing to a larger unit? You need fewer of them, so divide.
Multiplying or dividing by 10, 100 or 1,000 simply moves the decimal point 1, 2 or 3 places. For example, 2.5 km = 2.5 × 1,000 = 2,500 m, and 750 g = 750 ÷ 1,000 = 0.75 kg.
Worked examples: metric units
Example 1: Length
Question: The walk from school to the savannah is 1.35 km. How many metres is that?
Kilometres to metres is going to a smaller unit, so multiply.
1 km = 1,000 m, so 1.35 × 1,000 = 1,350.
Answer: 1,350 m.
Example 2: Mass
Question: A baker uses 750 g of flour from a 2 kg bag. How much flour is left, in kilograms?
Change 750 g to kilograms: 750 ÷ 1,000 = 0.75 kg.
Subtract: 2 − 0.75 = 1.25.
Answer: 1.25 kg (or 1,250 g). Always change to the same unit before you add or subtract.
Example 3: Capacity
Question: A 2 litre bottle of sorrel is shared equally into 8 glasses. How many millilitres go into each glass?
2 l = 2 × 1,000 = 2,000 ml.
2,000 ÷ 8 = 250.
Answer: 250 ml in each glass.
Metric and imperial: the unitary method
Imperial units (inches, feet, miles, pounds, gallons) are still used in the United States and you will still hear them at home, for example a person’s height in feet and inches, or a board’s length at the hardware. To change between the two systems we use a conversion fact:
Imperial
Metric
1 inch
2.54 cm (exactly)
1 foot (12 inches)
30.48 cm (about 30 cm)
5 miles
≈ 8 km
2.2 lb (pounds)
≈ 1 kg
1 gallon (UK)
≈ 4.5 litres
The sign ≈ means “is approximately equal to”. Answers found from an approximate fact are approximate too.
The unitary method uses the conversion fact in two steps:
Find one unit. Divide to find what 1 unit is worth.
Multiply. Multiply by the number of units you need.
Example 4: Kilometres to miles
Question: A road race is 20 km long. About how many miles is that? Use 8 km ≈ 5 miles.
One unit: 1 km ≈ 5 ÷ 8 = 0.625 mile.
Multiply: 20 km ≈ 20 × 0.625 = 12.5 miles.
Answer: about 12.5 miles.
Example 5: Pounds to kilograms
Question: A recipe from an American cookbook needs 10 lb of potatoes. About how many kilograms is this? Use 2.2 lb ≈ 1 kg.
One unit: 1 lb ≈ 1 ÷ 2.2 kg.
Multiply: 10 lb ≈ 10 ÷ 2.2 = 4.545… kg.
Answer: about 4.55 kg (to 2 decimal places), or roughly 4.5 kg.
Example 6: Feet and inches to centimetres
Question: Ravi says he is 5 feet 4 inches tall. What is his height in centimetres?
Change to inches: 5 feet = 5 × 12 = 60 inches, so 5 feet 4 inches = 64 inches.
1 inch = 2.54 cm, so 64 × 2.54 = 162.56 cm.
Answer: 162.56 cm, or about 163 cm.
Temperature: Celsius and Fahrenheit do not use a simple “times” rule because they start at different zeros. To change °C to °F, multiply by 9, divide by 5, then add 32. So 30 °C = 30 × 9 ÷ 5 + 32 = 86 °F, a typical hot day in Port of Spain.
Converting units of area
Area is measured in square units. Look at Figure 2: a square 1 cm by 1 cm is also 10 mm by 10 mm, so it holds 10 × 10 = 100 small squares.
Figure 2 – A 1 cm by 1 cm square (enlarged) holds 10 rows of 10 small squares, each 1 mm by 1 mm. So 1 cm² = 100 mm², not 10 mm².
So to convert area units, take the length conversion and square it (multiply it by itself):
Length
Area
1 cm = 10 mm
1 cm² = 10 × 10 = 100 mm²
1 m = 100 cm
1 m² = 100 × 100 = 10,000 cm²
1 m = 1,000 mm
1 m² = 1,000,000 mm²
1 km = 1,000 m
1 km² = 1,000,000 m²
Land is often measured in hectares (ha). A hectare is a square 100 m by 100 m, so 1 ha = 10,000 m², and 1 km² = 100 ha. A football field is roughly 0.7 ha.
The same rule applies: to a smaller unit, multiply; to a larger unit, divide. Just remember the numbers are bigger than for length.
Example 7: m² to cm²
Question: A notice board has an area of 1.2 m². What is its area in cm²?
1 m² = 10,000 cm² and cm² is the smaller unit, so multiply.
1.2 × 10,000 = 12,000.
Answer: 12,000 cm².
Example 8: mm² to cm²
Question: A postage stamp has an area of 450 mm². Write this in cm².
1 cm² = 100 mm² and cm² is the larger unit, so divide.
450 ÷ 100 = 4.5.
Answer: 4.5 cm².
Example 9: Tiling a floor
Question: A gallery floor is 5 m by 4 m. Square tiles have sides of 50 cm. How many tiles cover the floor?
Area of floor: 5 × 4 = 20 m² = 20 × 10,000 = 200,000 cm².
Area of one tile: 50 × 50 = 2,500 cm².
Number of tiles: 200,000 ÷ 2,500 = 80.
Answer: 80 tiles. Check another way: 5 m = 500 cm holds 10 tiles and 4 m = 400 cm holds 8 tiles, and 10 × 8 = 80. ✓
TRY ITUnit converter
1. Metric units (including area)
Choose the quantity and the two units, type a number, then press Convert. The steps show whether to multiply or divide.
2. Metric and imperial (unitary method)
Choose a conversion and type a number. Watch the two steps: find one unit, then multiply.
Common mistakes
Multiplying when you should divide. Ask yourself: is the new unit smaller (more of them) or larger (fewer of them)?
Using length numbers for area. 1 m² is 10,000 cm², not 100 cm².
Mixing units. Change everything to the same unit before adding, subtracting or finding an area.
Forgetting that some facts are approximate. 5 miles ≈ 8 km is close, not exact, so write “about” in your answer.
Skipping the “one unit” step. In the unitary method, finding one unit first makes the multiplying easy to check.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Convert: (a) 3.4 km to metres (b) 275 cm to metres (c) 6 mm to centimetres
Show answer
(a) 3.4 × 1,000 = 3,400 m (b) 275 ÷ 100 = 2.75 m (c) 6 ÷ 10 = 0.6 cm
2. Convert: (a) 1.75 kg to grams (b) 450 ml to litres (c) 2.3 tonnes to kilograms
Show answer
(a) 1.75 × 1,000 = 1,750 g (b) 450 ÷ 1,000 = 0.45 l (c) 2.3 × 1,000 = 2,300 kg
3. A bottle of juice holds 1.5 litres. How many 250 ml cups can be filled from it?
Show answer
1.5 l = 1,500 ml, and 1,500 ÷ 250 = 6 cups.
4. Which of these is equal to 0.5 m²? A. 50 cm² B. 500 cm² C. 5,000 cm² D. 50,000 cm²
Show answer
C. 1 m² = 10,000 cm², so 0.5 m² = 0.5 × 10,000 = 5,000 cm².
5. Convert: (a) 4 cm² to mm² (b) 25,000 cm² to m² (c) 3 km² to m²
6. Use 5 miles ≈ 8 km to convert: (a) 15 miles to kilometres (b) 40 km to miles
Show answer
(a) 1 mile ≈ 8 ÷ 5 = 1.6 km, so 15 miles ≈ 15 × 1.6 = 24 km. (b) 1 km ≈ 5 ÷ 8 = 0.625 mile, so 40 km ≈ 40 × 0.625 = 25 miles.
7. A bag of rice has a mass of 5 kg. Using 1 kg ≈ 2.2 lb, about how many pounds is this?
Show answer
5 × 2.2 = 11, so about 11 lb.
8. A television screen measures 32 inches across. Using 1 inch = 2.54 cm, how many centimetres is this?
Show answer
32 × 2.54 = 81.28 cm.
9. A classroom floor is a rectangle 8 m long and 6 m wide. (a) Find its area in m². (b) Write the area in cm². (c) How many square tiles of side 40 cm are needed to cover it?
Show answer
(a) 8 × 6 = 48 m² (b) 48 × 10,000 = 480,000 cm² (c) One tile is 40 × 40 = 1,600 cm², so 480,000 ÷ 1,600 = 300 tiles.
Quick check
1. To change metres to centimetres, do you multiply or divide, and by what number?
Show answer
Multiply by 100, because 1 m = 100 cm and centimetres are smaller.
2. Why is 1 m² not equal to 100 cm²?
Show answer
A square metre is 100 cm by 100 cm, so it holds 100 × 100 = 10,000 cm².
3. What are the two steps of the unitary method?
Show answer
First find the value of one unit, then multiply by the number of units you need.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 2, Strand 2.5 Measurement, Topic 2.5.1 Units of Measurement and Conversion of Units – Learning Outcomes 2.5.1.5 (converting units within the metric system), 2.5.1.6 (converting between systems of measure using the unitary method) and 2.5.1.7 (converting units of area).
Measuring is everywhere: a tailor measuring cloth for a school uniform, a vendor weighing tomatoes at the market, a nurse taking your temperature, or a teacher timing the 100 m on Sports Day. In Form 1 you learned the units of measurement and about mass and weight. In Lesson 4 you rounded and estimated. In this lesson you will choose the right instrument and unit for a job, read scales carefully, and say how accurate a measurement is.
Learning outcomes
By the end of this lesson you should be able to:
2.5.1.1 – Choose a suitable measuring instrument for a quantity.
2.5.1.2 – Choose the most suitable unit for measuring a quantity.
2.5.1.3 – Read and make sense of scales on measuring instruments.
2.5.1.4 – Measure quantities to a stated degree of accuracy, including time on the 24-hour clock.
Choosing a measuring instrument
Different jobs need different tools. For length alone there are several instruments, depending on how long the object is and how accurate you need to be.
Instrument
Measures, for example
Ruler or metre rule
Short lengths: a pencil, a page
Tape measure
Longer lengths: a room, your waist
Trundle wheel
Long distances: the school field
Vernier caliper
Small lengths: the width of a pipe
Micrometer screw gauge
Very small lengths: the thickness of a wire
Scale or balance
Mass: flour, a bag of rice, you
Measuring cylinder or jug
Capacity: medicine, water for cooking
Thermometer
Temperature: body or air temperature
Clock or stopwatch
Time: a race, a class period
Protractor
Angle: an angle in a triangle
A trundle wheel clicks once for every metre it rolls, so you count the clicks. A vernier caliper and a micrometer screw gauge are used in the science lab for lengths much smaller than a ruler can measure well.
Choosing the best unit
Choose a unit that gives a sensible number, not a huge one or a tiny decimal. You would give the length of your exercise book in centimetres, not kilometres!
Quantity
Units (small to large)
Example
Length
mm, cm, m, km
Coin thickness: mm. Door height: m.
Mass
mg, g, kg, tonne
A mango: g. A bag of rice: kg.
Capacity
ml, l
A spoon of medicine: ml. A bucket: l.
Temperature
°C
A hot day: about 32 °C.
Time
s, min, h
A sprint: s. A football match: min.
Reading a scale
Most scales have only some marks labelled. To read a scale:
Find two labelled marks next to each other, for example 30 and 40.
Count the spaces (not the lines) between them, for example 5.
Work out the value of one division: difference ÷ number of spaces. Here (40 − 30) ÷ 5 = 2.
Count the divisions from the labelled mark just below the reading, and add.
Figure 1 – Each small division on this ruler is 1 mm (0.1 cm). The bar starts at 2.0 cm and ends at 9.4 cm, so its length is 9.4 − 2.0 = 7.4 cm.
When you measure with a ruler, line the start of the object up with the 0 mark if you can. If you start at another mark, as in Figure 1, subtract: end reading − start reading.
Figure 2 – On this measuring cylinder there are 5 spaces between 30 ml and 40 ml, so each division is 2 ml. Read the bottom of the curved surface (the meniscus) with your eye level with it: 34 ml.
Water in a narrow tube curves at the top. This curve is the meniscus. Read the bottom of the curve, with your eye level with the surface. If you look from above or below, the reading will be wrong. This error is called parallax.
Figure 3 – This thermometer has 5 spaces between 30 °C and 40 °C, so each division is 2 °C. The liquid is 1 division above 30, so the temperature is 32 °C.
Degree of accuracy
No measurement is exact. The smallest division on the scale limits how accurate you can be. A ruler marked in millimetres measures to the nearest millimetre (0.1 cm). A kitchen scale marked every 20 g measures to the nearest 20 g, or perhaps the nearest 10 g if you judge halfway.
When a measurement is rounded, the true value can be up to half a unit either side. The bar in Figure 1 is 7.4 cm to the nearest 0.1 cm, so its true length is from 7.35 cm up to (but not including) 7.45 cm. This is the same idea as rounding in Lesson 4.
Value
Accurate to
True value lies
15 cm
nearest cm
14.5 cm to 15.5 cm
7.4 cm
nearest 0.1 cm
7.35 cm to 7.45 cm
250 g
nearest 10 g
245 g to 255 g
3 h
nearest hour
2.5 h to 3.5 h
Time and the 24-hour clock
Bus and ferry timetables, airline tickets and phones often use the 24-hour clock. It always uses four digits, and there is no a.m. or p.m.
For a.m. times, write the time with four digits: 7:05 a.m. is 07:05. Midnight is 00:00.
For p.m. times from 1:00 p.m. on, add 12 to the hours: 2:45 p.m. is 14:45. Noon is 12:00.
To find how long something takes, count on to the next hour, then on to the end time.
Worked examples
Example 1: Instrument and unit
Question: Choose an instrument and a unit for measuring (a) the thickness of a $1 coin (b) the length of the school corridor (c) the mass of a bag of rice.
A coin is very thin, so use a small, accurate instrument and a small unit.
A corridor is long, so use something that measures long lengths.
A bag of rice is heavy compared with a mango, so use kilograms.
Answer: (a) vernier caliper, mm (b) tape measure or trundle wheel, m (c) scale, kg
Example 2: Reading a kitchen scale
Question: A kitchen scale is labelled 0, 100, 200, 300 g with 5 spaces between labels. The pointer is 3 divisions past 200. What is the mass?
Value of one division: 100 ÷ 5 = 20 g.
Reading: 200 + 3 × 20 = 260 g.
Answer: 260 g.
Example 3: Reading a measuring cylinder
Question: Use Figure 2. What volume of water is in the cylinder?
There are 5 spaces between 30 ml and 40 ml, so each division is (40 − 30) ÷ 5 = 2 ml.
The bottom of the meniscus is 2 divisions above 30.
30 + 2 × 2 = 34.
Answer: 34 ml.
Example 4: Limits of accuracy
Question: A piece of ribbon is 12.6 cm long, measured to the nearest millimetre. Between which values does its true length lie?
The nearest millimetre is the nearest 0.1 cm. Half of 0.1 cm is 0.05 cm.
12.6 − 0.05 = 12.55 and 12.6 + 0.05 = 12.65.
Answer: from 12.55 cm up to 12.65 cm.
Example 5: A maxi-taxi journey
Question: A maxi-taxi leaves at 2:50 p.m. and arrives at 4:15 p.m. (a) Write both times in the 24-hour clock. (b) How long is the journey?
2:50 p.m. is 14:50 and 4:15 p.m. is 16:15.
From 14:50 to 15:00 is 10 minutes. From 15:00 to 16:15 is 1 hour 15 minutes.
Counting lines instead of spaces. Between 30 and 40 with 4 small lines in between there are 5 spaces, so each is worth 2, not 2.5.
Starting the ruler at its edge. Many rulers have a gap before 0. Start at the 0 mark, or subtract the start reading.
Reading the meniscus from above. Keep your eye level with the bottom of the curve.
Using a silly unit. The mass of a pencil in kilograms gives a tiny decimal; use grams.
24-hour clock slips. 15:00 is 3:00 p.m., not 5:00 p.m. Subtract 12 from the hours after noon.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Which instrument is best for measuring the diameter of a thin copper wire? A. A metre rule B. A tape measure C. A micrometer screw gauge D. A trundle wheel
Show answer
C. A micrometer screw gauge measures very small lengths, to about 0.01 mm.
2. Choose the best unit for each. (a) The mass of an egg (b) The length of a cricket pitch (c) The capacity of a teaspoon (d) The distance from Port of Spain to Arima
3. A scale is labelled 0, 10, 20, … with 5 equal spaces between each pair of labels. The pointer is 3 small divisions past 10. What is the reading?
Show answer
Each division is 10 ÷ 5 = 2. The reading is 10 + 3 × 2 = 16.
4. A measuring jug is labelled every 100 ml, with 4 spaces between labels. The juice is 2 divisions above the 300 ml mark. How much juice is in the jug?
Show answer
Each division is 100 ÷ 4 = 25 ml. The amount is 300 + 2 × 25 = 350 ml.
5. A leaf is placed on a ruler. One end is at 3.0 cm and the other end is at 11.7 cm. How long is the leaf, in cm and in mm?
Show answer
11.7 − 3.0 = 8.7 cm, which is 87 mm.
6. A bag of flour has a mass of 25 kg, correct to the nearest kilogram. Between which two values does its true mass lie?
Show answer
Between 24.5 kg and 25.5 kg (half of 1 kg either side).
7. On the thermometer in Figure 3, the liquid drops to 3 divisions above the 20 °C mark. What is the temperature now?
Show answer
Each division is 2 °C, so the temperature is 20 + 3 × 2 = 26 °C.
8. Write each time in the 24-hour clock. (a) 7:05 a.m. (b) 9:40 p.m. (c) Write 00:15 in the 12-hour clock.
Show answer
(a) 07:05 (b) 21:40 (c) 12:15 a.m. (a quarter past midnight)
9. On Sports Day, Kemba ran 100 m in 12.4 seconds, timed to the nearest tenth of a second. What are the shortest and longest times she could really have taken?
Show answer
From 12.35 s up to 12.45 s. The time is within half of 0.1 s, which is 0.05 s, of 12.4 s.
Quick check
1. On a ruler marked in millimetres, what is each small division worth in centimetres?
Show answer
0.1 cm (1 mm).
2. Where do you read the level of water in a measuring cylinder?
Show answer
At the bottom of the curved surface (the meniscus), with your eye level with it.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 2, Term 2, Strand 2.5 Measurement, Topic 2.5.1 Units of Measurement and Conversion of Units – Learning Outcomes 2.5.1.1 (choosing measuring instruments), 2.5.1.2 (choosing suitable units), 2.5.1.3 (reading and interpreting scales) and 2.5.1.4 (measuring to a given degree of accuracy).
Which is longer? Which is heavier? Today we measure and compare things.
Longer and shorter
To compare lengths, line things up at the same starting line. Then look at the other end.
✏️ pencil: ▬▬▬▬▬ 🖍️ crayon: ▬▬▬
The pencil is longer. The crayon is shorter. With three things we can say longest and shortest.
Measuring with paper clips
We can measure with paper clips, hand-spans, blocks or footsteps. Put them end to end:
in a straight line
no gaps
no overlaps
then count them
A book is 7 paper clips long. A pencil is 5 paper clips long. The book is 2 clips longer.
Heavier and lighter
A balance shows which side is heavier. The heavier side goes down. Photo: Nikodem Nijaki, CC BY-SA 3.0, via Wikimedia Commons.
Big things are not always heavier! A big balloon is very light. A small stone can be heavy.
TRY ITMeasure it!
Choose a game. Then tap your answer.
Let’s practise
Try each one. Then tap Show answer to check.
1. A ribbon is 6 clips long. A string is 9 clips long. Which is longer?
Show answer
The string.
2. Why must we leave no gaps between the clips?
Show answer
Gaps would make us count too few clips, so the answer would be wrong.
3. A table is 8 hand-spans long. A chair is 4 hand-spans long. How many more spans is the table?
Show answer
4 more hand-spans.
4. Which is heavier: a coconut or a feather?
Show answer
A coconut.
5. On a balance, the left side goes down. Which side is heavier?
Show answer
The left side.
6. Which is lighter: a big pillow or a small brick?
Show answer
The big pillow.
Quick check
1. Put in order, shortest first: a bus, a pencil, a car.
Show answer
Pencil, car, bus.
2. Is a bigger thing always heavier?
Show answer
No. A balloon is big but light.
3. What do we call it when both sides of a balance are level?
Show answer
They are as heavy as each other (they balance).
Maths words
Say these words with a grown-up. Can you use each one in a sentence?
long · longer · longest · short · shorter · shortest · tall · tallest · as long as · hand-span · heavy · heavier · light · lighter · as heavy as · weigh · balance
Try it at home
Measure your bed with hand-spans. Measure the table. Which is longer?
Line up your shoes and your family’s shoes from shortest to longest.
Hold a mango in one hand and a lime in the other. Which feels heavier? Check with a coat-hanger balance.
Make a pattern with long and short straws: long, short, long, short …
For parents and teachers
After this lesson, your child should be able to:
compare and order three or more objects by length (longest/shortest)
measure length using paper clips, hand-spans or blocks laid end to end with no gaps and no overlaps
compare masses using a balance and read pictures of balances
explain that a big object is not always heavier
Rice or beans in two identical cups hanging from a coat hanger makes a good home-made balance.
Perimeter is the total distance around the outside of a shape – the length of fence needed to go around a garden, or of ribbon to go around a picture frame. It is a length, so it is measured in units such as cm, m or km. Area is the amount of surface a shape covers – how much grass, paint or tiles you need. It is measured in square units such as cm2 (square centimetres) and m2 (square metres). In this lesson you will find the perimeter and area of squares, rectangles and triangles, and the perimeter of compound (L-shaped) figures.
To find the perimeter of any shape with straight sides, add the lengths of all the sides. For common shapes there are shortcuts:
Shape
Perimeter
Why
Square with side s
P = 4s
Four equal sides.
Rectangle with length l and width w
P = 2l + 2w = 2(l + w)
Two lengths and two widths.
Triangle with sides a, b, c
P = a + b + c
Add the three sides.
Any polygon
Add every side
Mark each side as you add it so you do not miss one.
Area
Area tells you how many unit squares fit inside a shape. A square that is 1 cm by 1 cm has an area of 1 cm2. On a grid you can find area by counting squares. For a shape with curved or uneven edges, estimate: count the whole squares, then count each square that is half or more covered as one, and ignore squares that are less than half covered.
Units: mm2 for tiny areas (a stamp), cm2 for a page, m2 for a room or a garden, and km2 for an island. (Trinidad and Tobago covers roughly 5,100 km2.) Large plots of land are often measured in hectares: 1 hectare = 100 m × 100 m = 10,000 m2.
Rectangle:A = l × w. A 6 cm by 4 cm rectangle holds 6 squares in each of 4 rows, so 24 cm2.
Square:A = s × s = s2.
Triangle:A = ½ × base × height, where the height is measured at a right angle (perpendicular) to the base. Any triangle is exactly half of a rectangle with the same base and height.
TRY ITPerimeter and area on a grid
Drag the sliders (or use the arrow keys) to change the rectangle. Watch how the perimeter and the area change – they are not the same thing!
Worked examples
Example 1: A rectangle
Question: Find the perimeter and area of a rectangle 9 cm long and 5 cm wide.
P = 2(l + w) = 2 × (9 + 5) = 2 × 14 = 28 cm.
A = l × w = 9 × 5 = 45 cm2.
Answer: Perimeter 28 cm, area 45 cm2.
Example 2: A square
Question: Find the perimeter and area of a square of side 7 m.
P = 4 × 7 = 28 m.
A = 7 × 7 = 49 m2.
Answer: Perimeter 28 m, area 49 m2. Notice: this square and the rectangle in Example 1 have the same perimeter number (28) but different areas.
Example 3: A right-angled triangle
Question: A triangle has sides 6 cm, 8 cm and 10 cm. The 6 cm and 8 cm sides meet at a right angle. Find its perimeter and area.
P = 6 + 8 + 10 = 24 cm.
The two sides that meet at the right angle are the base and the perpendicular height: base 8 cm, height 6 cm.
A = ½ × 8 × 6 = ½ × 48 = 24 cm2.
Answer: Perimeter 24 cm, area 24 cm2.
Example 4: Area of any triangle
Question: A triangle has a base of 12 cm and a perpendicular height of 5 cm. Find its area.
A = ½ × base × height = ½ × 12 × 5.
½ × 60 = 30.
Answer: 30 cm2. (The slanted sides are not needed for the area.)
Example 5: An L-shaped (compound) figure
Question: A garden is a 10 m by 8 m rectangle with a 4 m by 3 m rectangle cut out of one corner. Find its perimeter and area.
Work out the missing sides: going around, the sides are 10 m (bottom), 5 m (right, 8 − 3), 4 m (step in), 3 m (up), 6 m (top, 10 − 4) and 8 m (left).
Perimeter = 10 + 5 + 4 + 3 + 6 + 8 = 36 m. (Shortcut: cutting a corner out of a rectangle does not change its perimeter – it is still 2 × (10 + 8) = 36 m.)
Area method 2 (split): a 10 × 5 strip plus a 6 × 3 piece = 50 + 18 = 68 m2 ✔
Answer: Perimeter 36 m, area 68 m2.
Example 6: A real-life problem
Question: A rectangular garden is 12 m long and 7.5 m wide. Fencing costs $45 per metre, and one bag of grass seed covers 20 m2. Find the cost of the fence and the number of bags of seed needed.
You cannot buy half a bag, and 4 bags would not be enough, so round up.
Answer: The fence costs $1,755 and 5 bags of seed are needed.
Example 7: Finding a missing side
Question: A rectangle has an area of 54 cm2 and a length of 9 cm. Find its width.
A = l × w, so 54 = 9 × w.
w = 54 ÷ 9 = 6.
Answer: 6 cm.
Common mistakes
Wrong units. Perimeter is a length (cm); area is in square units (cm2). Writing “45 cm” for an area loses marks.
Adding only two sides. The perimeter of a 9 cm by 5 cm rectangle is 28 cm, not 14 cm.
Forgetting the ½ for a triangle, or using a slanted side instead of the perpendicular height.
Mixing units. Change everything to the same unit before calculating (for example 2 m by 50 cm → 200 cm by 50 cm).
Thinking a bigger perimeter always means a bigger area. A 9 × 1 rectangle has perimeter 20 but area only 9; a 5 × 5 square also has perimeter 20 but area 25.
Practice questions
Try these yourself first, then open each answer to check your working.
1. Find the perimeter and area of a rectangle 15 cm by 4 cm.
Show answer
Perimeter = 2 × (15 + 4) = 38 cm. Area = 15 × 4 = 60 cm2.
2. A square has a perimeter of 36 cm. Find its side and its area.
Show answer
Side = 36 ÷ 4 = 9 cm. Area = 9 × 9 = 81 cm2.
3. A triangle has a base of 10 m and a perpendicular height of 7 m. Find its area.
Show answer
½ × 10 × 7 = 35 m2.
4. Find the area of a rectangle 2 m long and 50 cm wide, in cm2.
Show answer
2 m = 200 cm, so 200 × 50 = 10,000 cm2. (That is 1 m2 – which shows that 1 m2 = 10,000 cm2, not 100!)
5. A floor is 5 m by 4 m. How many square tiles of side 50 cm are needed to cover it?
Show answer
Floor area = 20 m2; each tile = 0.5 × 0.5 = 0.25 m2; 20 ÷ 0.25 = 80 tiles. (Or: 10 tiles along the 5 m side × 8 along the 4 m side = 80.)
6. A 12 m by 9 m rectangle has a 5 m by 4 m rectangle cut from one corner. Find the perimeter and area of the shape that is left.
Show answer
Perimeter = 2 × (12 + 9) = 42 m (cutting a corner does not change it). Area = 108 − 20 = 88 m2.
7. Draw (or describe) three different rectangles with whole-number sides and a perimeter of 20 cm. Which has the biggest area?
Show answer
For example 9 × 1 (area 9 cm2), 6 × 4 (24 cm2) and 5 × 5 (25 cm2). The square has the biggest area.
8. A hectare is the area of a 100 m by 100 m square. How many square metres is that?
Show answer
100 × 100 = 10,000 m2.
Quick check
1. What is the difference between perimeter and area?
Show answer
Perimeter is the distance around a shape (a length); area is the surface inside it (square units).
2. Write the formula for the area of a triangle.
Show answer
Area = ½ × base × perpendicular height.
3. Which unit would you use for the area of a classroom floor?
Show answer
Square metres (m2).
4. A shape on a grid covers 9 whole squares and 4 squares that are more than half covered. Estimate its area in square units.
Curriculum link: This post is relevant to the Trinidad and Tobago National Lower Secondary (NCSE) Mathematics Curriculum – Form 1, Term 1, Strand 1.5 Measurement, Topic 1.5.3 Perimeter – Learning Outcomes 1.5.3.1 (identifying the perimeter of plane shapes), 1.5.3.2 (calculating perimeter) and 1.5.3.3 (solving perimeter problems to a stated degree of accuracy); and Topic 1.5.4 Area – Learning Outcomes 1.5.4.1 (the concept of area), 1.5.4.2 (units for area), 1.5.4.3 (measuring and estimating surface area) and 1.5.4.4 (the area of triangles, squares and rectangles).
