A second picture

Next to the photo of the aerial, Erklärus showed Liri a second picture: a folded-up solar-sail panel. It wasn’t rectangular, but became narrower towards the top – two sides ran parallel, the other two slanted towards each other.

Liri: “Why isn’t the panel simply rectangular, like most solar sails?”
Erklärus: “It is exactly this slanted shape that can be folded more compactly when it is closed up, like a fan. This shape is called a trapezium: only two sides are parallel to each other, the other two are allowed to slope.”

🔷 Definition – what makes a trapezium?

Erklärus: “A trapezium is a four-sided shape with exactly one pair of parallel sides. These two sides are called the parallel sides – the longer one is usually called a, the shorter one c. The other two, non-parallel sides are called legs – b and d.”
Liri: “And the distance between the two parallel sides?”
Erklärus: “That is the height h – it always stands at a right angle to the two parallel sides, no matter how slanted the legs are.”
A B C D a (long parallel side) c (short parallel side) d b h
Labels in the trapezium
Parallel side a
The longer of the two parallel sides
Parallel side c
The shorter of the two parallel sides
Legs b, d
The two non-parallel sides
Height h
Perpendicular distance between a and c
Liri: “So a rectangle is really also a trapezium, just with parallel sides of equal length?”
Erklärus: “Well spotted! A rectangle is a special case of the trapezium.”

⚖️ The isosceles trapezium – the special shape

Liri: “Is there a special shape here too, like the square for the rectangle?”
Erklärus: “Very observant! There is the isosceles trapezium. Here the two legs b and d are the same length. That creates perfect symmetry – a line of symmetry runs exactly through the middle.”
Liri: “And what does that mean for the angles?”
Erklärus: “Because of the symmetry, the angles at each parallel side are the same size: α = β at the long side. The two diagonals are also the same length – just like in a rectangle.”
A B C D α β α = β
Properties of the isosceles trapezium
• The legs b and d are the same length
• The base angles are equal in pairs: α = β
• The two diagonals are the same length
• It has a line of symmetry at right angles to the parallel sides

📏 The perimeter

As with every four-sided shape: add up all four sides.

P = a + b + c + d
Sides: 8, 4, 5, 5 cm. Perimeter?

📐 The area – the trick with the mean

Liri: “And the area? The trapezium has two parallel sides of different lengths – how do you multiply that?”
Erklärus: “A lovely thought. Imagine you take two identical trapeziums and turn the second one round by 180°. If you put them together, you get a parallelogram – with the base (a + c) and the same height h.”
Liri: “And the area of this parallelogram is (a + c) times h. But that is two trapeziums together!”
Erklärus: “Well spotted. So we divide by 2 once more: A = (a + c) × h ÷ 2. You can also remember it like this: you take the mean of the two parallel sides and multiply it by the height.”
h c a c a Parallelogram: base = a + c
Formula – area of the trapezium
A = (a + c) × h ÷ 2
a, c = parallel sides · h = height (perpendicular distance between a and c)
Erklärus: “This formula even works for the rectangle! If a = c, then (a+c)/2 = a, and you get A = a × h again – the old rectangle formula. So the rectangle is a special case of the trapezium.”
Bottom side 10, top side 6, height 4 cm. Area?

🧮 Liri’s trapezium calculator – interactive

Enter the two parallel sides and the height. The calculator draws the trapezium and works out the area.

🔷 Calculate and draw a trapezium
🌌 Trapeziums in the universe

Many rocket fins and aerial dishes have a trapezium-shaped basic form, to save material and still be stable.
Solar sails are sometimes cut in the shape of a trapezium too, to fit the shape of a spaceship as well as possible.

❌ Common mistake

In the area formula people often forget that ONLY the two parallel sides (a and c) are added – not all four sides!

Tip: A = (a+c)÷2 × h – the slanted sides b and d don’t appear in this formula at all.

📏 Exercise 1 – Perimeter
Sides: 6, 3, 4, 4 cm. Perimeter?
📐 Exercise 2 – Area
Bottom side 12, top side 8, height 5 cm. Area?
🔷 Exercise 3
Bottom side 14, top side 6, height 3 cm. Area?

A sail, understood anew

Liri worked out the area of the solar-sail panel – and then she stopped short.

Liri: “This slanted shape… it seems familiar. The solar panels at my home looked like that – wide at the front, narrow at the back, so that they could be folded up compactly when docking.”
Erklärus: “Then this panel is perhaps more than just a shape on your map. If you recognise WHERE such trapezium sails are built, you will get closer to the answer of where your home is.”
Liri: “Even slanted shapes can be worked out if you know the right trick – and sometimes they even lead me closer to home.”
“Trapezium: two parallel sides (a, c). Perimeter: all four sides added. Area: A=(a+c)÷2×h – mean of the parallel sides times height.”
What you learned today
A = (a+c) ÷ 2 × h
Only the parallel sides a and c count for the area
🌟

Chapter 37 mastered!

Liri can now work out the perimeter and area of every trapezium.

Erklärus’s notebook – page 37:
“Trapezium: exactly two parallel sides (a and c). Perimeter: all four sides added. Area: A=(a+c)÷2×h – the mean of the parallel sides times height.”
← Back to overview 37 – The Trapezium