Area of a square and a rectangle · البرمجة مع مارتن

Area as a count of little squares: S = a × b and why it differs from perimeter.

How many little squares fit inside

How many tiles do you need for a floor? How much paper for a poster? How much lawn is on a pitch? That is all a question about area — the size of a surface.

Picture area like this: pave the shape with little squares 1 cm × 1 cm and count them. How many little squares fitted, that is the area in square centimetres (cm²).

With a rectangle you do not have to count them one by one. A rectangle 5 cm × 3 cm has 3 rows of 5 little squares — a times table: 5 × 3 = 15 little squares. From that the formula: S = a × b (side times side). A square has both sides the same, so S = a × a.

Watch, area is not perimeter! Perimeter is the length of a fence around a plot (you add the sides, it comes out in cm). Area is the grass inside the fence (you multiply the sides, it comes out in cm²). Two different questions, two different calculations, two different units. Today we split them apart for good.

Filip is tiling a bathroom with dad. The wall is 4 metres long and 2 metres high. “How many tiles do we buy? They are metre tiles,” dad asks. Filip counts out loud: “Four plus two… six?” Dad draws a grid on the wall in pencil: two rows, four tiles in each. “Count.” Filip counts the squares: eight. “Ah — rows times columns. Four times two!” Exactly. By adding they would buy six tiles and two would be missing. Area is multiplied because little squares make a grid — rows times columns, like in a table.

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The unit gives away what you are counting: perimeter comes out in cm (a line), area in cm² (a surface). If with area you only get cm, multiplying got lost somewhere.

Formulas and how to handle them

Rectangle: S = a × b. Sides a, b are length and width. A rectangle 7 cm × 4 cm: S = 7 × 4 = 28 cm².

Square: S = a × a. A square with side 6 cm: S = 6 × 6 = 36 cm².

Method for a word problem:

Find both sides and check they are in the same unit. A metre and a centimetre must not be multiplied — convert first (1 m = 100 cm).
Multiply: S = a × b.
Write the unit to the power of two: cm², m². We read “square centimetre” — it is a little square 1 cm × 1 cm.

The other way — I know the area, I look for a side: a rectangle has area 24 cm² and one side 6 cm. The other side = 24 : 6 = 4 cm. Multiplying backwards is dividing.

And perimeter for comparison: o = 2 × (a + b) — a path all the way round, two lengths and two widths. A rectangle 7 × 4: perimeter = 2 × 11 = 22 cm, area = 28 cm². Different numbers, different units, different questions.

A tricky thing at the end: two rectangles can have the same perimeter and a different area. A rectangle 1 × 5 and 3 × 3 both have perimeter 12 cm — but areas 5 cm² and 9 cm². The same fence, different grass!

Area and perimeter of a rectangle 8 cm × 5 cm

Area: S = a × b = 8 × 5 = 40 cm². 40 little squares a centimetre by a centimetre fit there (5 rows of 8).

Perimeter: o = 2 × (8 + 5) = 2 × 13 = 26 cm. All the way round it is 26 centimetres of line.

Notice the difference in units: area cm², perimeter cm. Two numbers about one rectangle, each answers a different question.

How many square metres does a square garden have?

The garden is a square with side 9 m.

S = a × a = 9 × 9 = 81 m².

A picture check: a grid 9 × 9 of metre squares — nine rows of nine — that is the nine times table.

And if you asked about the fence? Perimeter = 4 × 9 = 36 m. Eighty-one square metres of grass, but only thirty-six metres of netting. Area and perimeter each live their own life.

I know the area, I look for a side

A poster has area 48 cm² (all right, a small poster!) and width 6 cm. How high is it?

I know: S = a × b, so 48 = 6 × height.

The other way: height = 48 : 6 = 8 cm.

Check: 6 × 8 = 48 ✓.

You need this reverse often: you know the area of a room and one wall, you look for the other. The formula works both ways — forwards you multiply, backwards you divide.

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Do not multiply metres with centimetres. A wall 4 m × 50 cm is not 200 of anything — first unify the units: 4 m = 400 cm, then 400 × 50 = 20 000 cm². You may only multiply the same units, otherwise a number with no sense comes out.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. البداية with ten. When it goes well, add more.

Work out the area of a rectangle: side times side. Write the answer as a number (in cm²).

Practise the questions

On paper

Squared paper is your best friend today.

Draw on squared paper a rectangle 6 × 4 little squares. Work out the area with the formula and check by counting the little squares. Then work out the perimeter too.
Work out the areas: a rectangle 12 cm × 7 cm, a square with side 11 cm, a rectangle 2 m × 150 cm (watch the units!).
A rectangular table has area 72 dm² and length 9 dm. What is its width? Do a check.
Draw two different rectangles that both have perimeter 16 cm. Work out their areas. Which has the bigger area — and can you find a rectangle with perimeter 16 and an even bigger area?

Now you 💪

I can work out the area of a rectangle and a square and write the right unit (cm², m²).
I can explain the difference between area and perimeter — for example on a lawn and a fence.
I can work out a side from the area by dividing.

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Done when: You work out the area of a square and a rectangle with the formula, you do not mix it up with perimeter, and before multiplying you unify the units.

What to take from this lesson

Area = the number of unit little squares inside. Rectangle: S = a × b, square: S = a × a.
Area comes out in square units: cm², m².
Perimeter is a line around (adding), area is a surface inside (multiplying).
Before calculating, unify the units of both sides.
You reverse the formula by dividing: side = area : the other side.