Area of a square and a rectangle · Lập trình với Martin

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. Bắt đầu 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.