A field on the map

Liri had marked a rectangular area on her star map – a region that she still had to search.

Liri: “How big is this field, actually? And how far is it once all the way round?”
Erklärus: “Two different questions, two different calculations: the perimeter – once all the way round – and the area – how much space there is inside.”

📐 The rectangle – four right angles, two pairs of sides

Erklärus: “A rectangle is a flat shape with four sides and four right angles. The four corners are called A, B, C, D – we always go round clockwise. We call the longer side the length a, the shorter side the width b.”
Liri: “And what about the opposite sides?”
Erklärus: “That is the special thing: opposite sides are always the same length and parallel to each other. So a rectangle has two pairs of parallel sides.”
A B C D a (length) b (width) a (length) b
Properties of the rectangle
Sides
4 sides: 2× length a, 2× width b
Opposite sides equal in length and parallel
Angles
4 right angles (90° each)
All four corners are right angles
Diagonals
2 diagonals: equal in length
They bisect each other at the centre

📏 The perimeter

The perimeter is the distance once all the way round. For a square (all 4 sides equal in length), one side times 4 is enough. For a rectangle you add the length and the width and double it.

Erklärus: “Imagine you walk once all the way round the outside of the field – you measure all four sides together. That is the perimeter.”
Liri: “So twice the length and twice the width?”
Erklärus: “Exactly right. And because people like to shorten things: twice the sum of the length and the width.”
P(square) = 4 × a
P(rectangle) = 2 × (a + b)
Square with side 6 cm – perimeter?

🟩 The area

The area shows how much space there is inside the shape – measured in square centimetres (cm²) or square metres (m²).

A(square) = a × a
A(rectangle) = a × b
Erklärus: “Imagine a tile – 5 cm long and 4 cm wide. If we cover it with small squares that are exactly 1×1 cm in size, how many fit in?”
12345 678910 1112131415 1617181920 5 cm 4 cm
Liri: “I count 5 squares in every row… and there are 4 rows. So 5+5+5+5 – that’s 20 squares!”
Erklärus: “Exactly 20. And 5 added four times is the same as 5 times 4 – that is what multiplication means: adding the same number several times. That is why for a rectangle we simply multiply length times width.”
Rectangle: 5 cm × 3 cm – area?

🧮 Liri’s shape calculator – interactive

Enter the sides and watch how the perimeter and area change live:

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🟨 The square – the special case of the rectangle

Erklärus: “The square isn’t a strange creature – it is a rectangle with one special property: all four sides are the same length. There is only a single side length left – we call it a.”
Liri: “So every square is also a rectangle, but not every rectangle is a square?”
Erklärus: “Perfectly put! The square is a special case. It inherits all the properties of the rectangle – four right angles, parallel opposite sides, equal diagonals – and adds: all four sides the same length.”
A B C D a a a a
Properties of the square
Sides
4 sides: all the same length (a)
a = b – that is what makes it a square
Angles
4 right angles (90° each)
Inherited from the rectangle
Diagonals
Equal in length, bisect each other
Perpendicular to each other and bisect the corner angles
🌌 Areas in the universe

The solar sails of space probes are often rectangular, to catch as much sunlight as possible in a small space.
A solar panel of 4 m × 2 m has an area of 8 m² – the bigger the area, the more energy.
Without calculating area, engineers couldn’t plan how much energy a satellite can collect.

❌ Common mistake

Perimeter and area are often mixed up: the perimeter is given in cm (a simple unit of length), the area in cm² (a square unit)!

Tip: perimeter = walking once all the way round. Area = how many squares fit inside.

📏 Exercise 1 – Perimeter
Rectangle: 8 cm long, 3 cm wide. What is the perimeter?
🟩 Exercise 2 – Area
Square with side 7 cm. What is the area?
🛰️ Exercise 3 – Universe
A solar panel is 6 m long and 2 m wide. What is the area?

The field is measured

Liri wrote the perimeter and area of her marked field onto the star map – now she knew exactly how big the region was that she still had to search.

Liri: “Now I don’t only know where my home might be, but also how big the region is that I have to search.”
“Perimeter: once all the way round. Square: 4×a. Rectangle: 2×(a+b). Area: how much space there is inside. Square: a×a. Rectangle: a×b.”
What you learned today
P = 2×(a+b)  ·  A = a×b
Perimeter in cm/m · Area in cm²/m²
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Chapter 23 mastered!

Liri can now work out the perimeter and area of squares and rectangles.

Erklärus’s notebook – page 23:
“Perimeter: P(square)=4×a, P(rectangle)=2×(a+b), unit cm/m. Area: A(square)=a×a, A(rectangle)=a×b, unit cm²/m². Never mix them up!”
← Back to overview 23 – Square & Rectangle