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Perimeter and area of 2D polygons
Rectangles, triangles, parallelograms and compound rectilinear shapes.
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The written summary and downloads below are complete and ready to use.
The perimeter of a shape is the distance all the way round its edge, like a fence round a field. The area is the amount of flat space inside it, like the carpet on a floor. In this lesson you will learn how to:
- find the perimeter of a shape by adding all its sides
- find an area by counting squares, including half squares
- find the area of a rectangle: length × width, and work backwards to a missing side
- find the area of a triangle: half of a rectangle
- find the area of a parallelogram: cut it and slide it into a rectangle
- find missing sides, perimeters and areas of compound shapes made of rectangles
Dark blue: whole numbers
Red dot: the decimal point
Blue: digits after the point
1. Perimeter: all the way round
Key idea
The perimeter is the total length of all the sides of a shape. Start at a corner, go all the way
round, and add up every side. A perimeter is a length, so it is measured in mm, cm, m or km.
Example 1: each square is 1 cm wide. What is the perimeter?
- Start at the bottom left corner and count the edges of the squares as you go round:4 + 2 + 2 + 1 + 2 + 3
- Add them up:= 14
AnswerThe perimeter is 14 cm.
Example 2: what is the perimeter of this rectangle?
- Opposite sides of a rectangle are the same length, so the top is 8 cm and the left side is 3 cm:8 + 3 + 8 + 3
- Add them up:= 22
AnswerThe perimeter is 22 cm.
Example 3: a regular hexagon has sides of 5 cm. What is its perimeter?
- A regular shape has all its sides the same length. The little ticks show this. A hexagon has 6 sides:6 × 5 = 30
AnswerThe perimeter is 30 cm.
Watch out
Count every side, even the ones without a label. A rectangle has 4 sides, so 8 + 3 = 11 cm only gets
you halfway round.
The ant’s walk
How to play
- Work out the perimeter and type it in. Then the ant walks all the way round the edge, adding up each side as it goes, so you can check.
- Two with no slips moves you up a level. Level 1 has shapes on squared paper, level 2 has shapes with labelled sides, and on level 3 some sides are missing, so you work them out first.
2. Area: counting squares
Key idea
The area is the amount of space inside a shape. We measure it in squares. A square 1 cm long and
1 cm wide has an area of 1 square centimetre, written 1 cm². Bigger areas, like a
classroom floor, are measured in square metres (m²).
Example 1: each square is 1 cm². What is the area?
- Count the squares inside: 8 in the bottom part, and 2 more on top:8 + 2 = 10
AnswerThe area is 10 cm². (It is the same
shape as in section 1: its perimeter is 14 cm, but its area is 10 cm².)
Example 2: each square is 1 cm². What is the area?
- Count the whole squares:7
- Two half squares make one whole square:2 halves = 1
- Add them:7 + 1 = 8
AnswerThe area is 8 cm².
Watch out
A perimeter is a length, in cm. An area is a number of squares, in cm². Always write
the right unit with your answer.
Count the squares
How to play
- Tap the squares inside the shape to count them. Each one gets its number. Then type the area.
- Two with no slips moves you up a level. Level 1 has whole squares, level 2 has half squares too, and on level 3 you build your own shape with the area and perimeter asked for.
3. Rectangles: length × width
Key idea
A rectangle is made of rows of squares. The length tells you how many squares are in each row, and
the width tells you how many rows there are. So area = length × width. For the
perimeter, add the four sides: length + width + length + width.
Example 1: a rectangle is 7 cm long and 4 cm wide. Find its area and its perimeter.
- Area: 4 rows of 7 squares:7 × 4 = 28
- Perimeter: add the four sides:7 + 4 + 7 + 4 = 22
AnswerArea 28 cm², perimeter 22 cm.
Example 2: a rectangle has an area of 36 m². It is 9 m long. How wide is it?
- Length × width = area, so ask: 9 × what = 36? Divide:36 ÷ 9 = 4
AnswerIt is 4 m wide.
Example 3: a square has a perimeter of 20 cm. What is its area?
- A square has 4 equal sides:20 ÷ 4 = 5
- Area = length × width:5 × 5 = 25
AnswerThe area is 25 cm².
Watch out
Area multiplies, perimeter adds. Two rectangles can have the same area but different perimeters:
a 6 by 2 rectangle and a 4 by 3 rectangle both have an area of 12 cm², but their perimeters are 16 cm
and 14 cm.
Rectangle builder
How to play
- Drag the red corner (or use the − and + buttons) to change the rectangle. Its area and perimeter change as you go. Press Check when it matches the challenge.
- Two with no slips moves you up a level. Level 1 gives you an area to make, level 2 gives you an area and a perimeter, and level 3 asks for the smallest or biggest perimeter.
4. Triangles: half a rectangle
Key idea
Every triangle is half of a rectangle with the same base and height. So
area of a triangle = base × height ÷ 2. The height is measured straight up from
the base, at a right angle to it (the little square shows this). It is not the length of a sloping side.
Example 1: find the area of this right-angled triangle.
- The whole rectangle would be:8 × 5 = 40
- The triangle is half of it:40 ÷ 2 = 20
AnswerThe area is 20 cm².
Example 2: find the area of this triangle.
- Base × height (the dashed line, not the 10 cm side):12 × 8 = 96
- Halve it:96 ÷ 2 = 48
AnswerThe area is 48 cm². The 10 cm side is
not needed.
Watch out
The two slips that cost the most marks: forgetting to halve, and using the sloping side
instead of the height.
Slide the top
Drag the red top corner from side to side. Watch the area.
What to notice
- The base and the height stay the same, so the area never changes, however much the triangle leans.
- Show the rectangle: the dashed height cuts it into two smaller rectangles, and the triangle is exactly half of each one. The pale parts outside match the parts inside, piece for piece.
5. Parallelograms: cut and slide
Key idea
A parallelogram is a leaning rectangle: its opposite sides are parallel and the same length. Cut the
triangle off one end and slide it to the other end, and you get a rectangle with the same base and height.
So area of a parallelogram = base × height. There is no halving this time.
Example: find the area of this parallelogram.
- Base × height (the dashed line, not the 5 cm side):9 × 4 = 36
AnswerThe area is 36 cm².
Why it works
Cut along the dashed height and move triangle A to the other end. Nothing is lost and nothing is added, so the area stays the same: a rectangle 9 cm by 4 cm.
Watch out
Do not halve a parallelogram: that is only for triangles. And do not multiply the two sloping sides:
9 × 5 = 45 is wrong. Use the height, at right angles to the base.
Cut and slide
How to play
- Follow the steps and type each number. Press Cut and slide at any time to turn the parallelogram into a rectangle and back.
- Two with no slips moves you up a level. Level 1 is on squared paper, level 2 has a sloping side to ignore, and level 3 works backwards from the area.
6. Compound shapes: missing sides and perimeter
Key idea
A compound shape is made of rectangles joined together. When every corner is a right angle, the
sides going across the top add up to the side across the bottom, and the sides going up on
one side add up to the side on the other. Use this to find missing sides.
Example: find the two missing sides, then the perimeter.
- Across (red): the top parts make 12 cm, and one of them is 7 cm:12 − 7 = 5
- Up (blue): the right-hand parts make 9 cm, and one of them is 4 cm:9 − 4 = 5
- Now add all six sides:12 + 4 + 5 + 5 + 7 + 9 = 42
AnswerThe missing sides are 5 cm and 5 cm. The
perimeter is 42 cm.
Watch out
An L shape has 6 sides. Count them as you add, and tick each one off. A quick check: the red sides
on top (7 + 5) should make the bottom (12).
Missing side detective
How to play
- Tap a side marked ?. The sides that help you find it light up. Type its length. When every side is known, find the perimeter.
- Two with no slips moves you up a level. Level 1 has one missing side, level 2 has two, and level 3 has bigger shapes with more steps.
7. Compound shapes: area
Key idea
There are two ways. Split it into rectangles, find the area of each one, and add. Or take it
away: find the area of a big rectangle round the whole shape, and take away the piece that is
missing. Both ways give the same answer, so you can use one to check the other.
Example 1: split it
- The bottom rectangle:12 × 4 = 48
- The top rectangle is 7 cm across and 9 − 4 = 5 cm tall:7 × 5 = 35
- Add them:48 + 35 = 83
AnswerThe area is 83 cm².
Example 2: take it away
- The big rectangle:12 × 9 = 108
- The missing corner is 5 by 5:5 × 5 = 25
- Take it away:108 − 25 = 83
AnswerThe area is 83 cm² again.
Example 3: a path goes round a lawn. What is the area of the path?
- The whole thing, path and lawn:14 × 9 = 126
- The lawn:10 × 5 = 50
- Take the lawn away:126 − 50 = 76
AnswerThe path has an area of 76 m².
Watch out
Do not multiply all the numbers you can see. 12 × 9 is the big rectangle, not the L shape. Find
each rectangle’s own length and width first.
Split it or take it away
How to play
- Pick a way: cut across, cut down, or take away. The shape splits into coloured rectangles. Find the area of each one, then the whole shape.
- Two with no slips moves you up a level. On level 1 every side is labelled, on level 2 you find a missing side first, and level 3 has paths, frames and bigger shapes.
8. Perimeter and area problems
Key idea
First decide: is the question about the edge (a fence, a frame, a border, walking round: perimeter)
or the inside (carpet, grass, paint, tiles: area)? Then draw a quick sketch and label it.
Example 1: a garden is 9 m long and 6 m wide. Fencing costs £5 a metre. How much does it cost to fence all the way round?
- A fence goes round the edge, so find the perimeter:9 + 6 + 9 + 6 = 30
- 30 metres at £5 each:30 × 5 = 150
Answer£150
Example 2: a wall is 8 m long and 3 m high. One tin of paint covers 6 m². How many tins are needed?
- Paint covers the inside, so find the area:8 × 3 = 24
- How many 6s make 24?24 ÷ 6 = 4
Answer4 tins
Watch out
Read the question again at the end. Did it ask for the area, the perimeter, a missing side, or a cost?
Perimeter and area workshop
How it works
- Follow the steps and type each number. The . key is the decimal point.
- Two with no slips moves you up a level. Level 1 is rectangles, level 2 works backwards, level 3 is triangles and parallelograms, level 4 has fences, paint and tiles, and level 5 has puzzles that take two or three steps.
9. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Move up when you are getting most of them right.
The mistakes that cost marks
- Mixing up area and perimeter. Adding the sides when the question asks for the area. area is the inside (multiply for a rectangle); perimeter is the edge (add every side).
- Missing sides off the perimeter. Saying a 8 cm by 3 cm rectangle has a perimeter of 11 cm. go all the way round: 8 + 3 + 8 + 3 = 22 cm.
- The wrong unit. Writing an area in cm, or a perimeter in cm². perimeter in cm or m; area in cm² or m².
- Forgetting to halve a triangle. a triangle is half a rectangle: base × height ÷ 2.
- Using a sloping side as the height. the height is at right angles to the base: look for the little square.
- Halving a parallelogram. a parallelogram is base × height, with no halving.
- Multiplying the two longest sides of an L shape. split it into rectangles, or take away the missing piece.
- Leaving out a side of a compound shape. count the sides first (an L shape has 6) and tick each one off as you add it.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 6.3.
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