A picture that wakes something up

Erklärus showed Liri an old photo of a satellite dish – its framework was made entirely of little triangles, like a net of struts. Liri sensed straight away: she knew this.

Liri: “Why don’t they simply build something like that out of squares? It would look tidier.”
Erklärus: “Because a square can bend without its side lengths changing – a triangle can’t. That is why real engineers build satellites, space stations and aerial masts out of triangle lattices. It is the most stable shape there is. Let’s find out why – perhaps it will help you remember more.”

🔺 Basic terms – sides, vertices, angles

Erklärus: “A triangle has three vertices – we call them A, B and C. The opposite side is always named after the lower-case letter of the vertex: side a lies opposite vertex A, side b opposite vertex B, side c opposite vertex C.”
Liri: “An elegant rule – the letter stays, only capital and small letters swap.”
Erklärus: “Exactly. And the three interior angles are called α (alpha) at vertex A, β (beta) at vertex B and γ (gamma) at vertex C.”
A B C α β γ a b c
Labels in triangle ABC
Vertices
A, B, C
Sides
a (opposite A), b (opposite B), c (opposite C)
Angles
α (at A), β (at B), γ (at C)

📐 The angle sum – always 180°

No matter what a triangle looks like – its three interior angles ALWAYS add up to exactly 180°.

α + β + γ = 180°
Erklärus: “If you know two angles, you can always work out the third – simply subtract from 180°.”
Two angles are 70° and 60°. How big is the third?

📊 Types of triangle by their angles

Erklärus: “We can divide triangles into three families according to their angles.”
Spitzwinklig
All 3 angles < 90°
Rechtwinklig
One angle = 90°
Stumpfwinklig
One angle > 90°
Triangles by angles
Acute-angled
All three angles smaller than 90°
Right-angled
Exactly one angle = 90°
Obtuse-angled
Exactly one angle bigger than 90°

📏 Types of triangle by their sides

Erklärus: “The second classification looks at the sides: how many sides are the same length?”
Gleichseitig
All 3 sides equal
All angles 60°
Gleichschenklig
2 sides the same length
2 base angles equal
Ungleichseitig
All 3 sides different lengths
Triangles by sides
Equilateral
a = b = c → all angles 60°
Isosceles
Two sides equal → two base angles equal
Scalene
All sides different lengths

📏 Perimeter and area

The perimeter is simple: all three sides added. P = a + b + c. No tricks.

Liri: “And the area? That surely isn’t as simple as with the rectangle.”
Erklärus: “Look – every triangle is exactly half of a parallelogram, and every parallelogram can be turned into a rectangle. Imagine a rectangle with width equal to the base and height h. The triangle with the same base and height has exactly half the area.”
Liri: “So: base times height divided by two!”
base h A = base·h/2 Rectangle: base × h
Formulas – triangle
P = a + b + c
A = (base × h) ÷ 2
base = base line · h = height on the base · P = perimeter · A = area
Erklärus: “Important: the height h stands at right angles to the base. You can choose any of the three sides as the base – the area always stays the same.”
Triangle: base 8 cm, height 5 cm. Area?

🧮 Liri’s triangle calculator – interactive

Enter the three sides. The calculator draws the triangle and works out the perimeter, area and angles.

🔺 Calculate and draw a triangle
🌌 Triangles in the universe

Satellite wings and aerial supports are often built in triangle shapes – triangles are extremely stable and don’t deform easily.
Astronomers use triangle calculations (triangulation) to determine the distances to faraway stars.

❌ Common mistake

In the area formula the division by 2 is often forgotten: base × h alone is the area of the RECTANGLE, not of the triangle!

Tip: triangle = rectangle ÷ 2. The formula is A = base × h ÷ 2, not A = base × h.

📐 Exercise 1 – Angle sum
Two angles are 90° and 45°. How big is the third?
📏 Exercise 2 – Perimeter
Triangle with sides 5, 6 and 7 cm. What is the perimeter?
🔺 Exercise 3 – Area
Triangle: base 10 cm, height 4 cm. What is the area?

A stable way of building

Liri now understood why the old aerial framework was made of nothing but triangles – no other shape was so stable and simple at the same time. And then she remembered something.

Liri: “Wait – I know triangular struts like that! The aerials and wings at home were built in exactly the same way, with real triangles, so that nothing wobbles.”
Erklärus: “That is a real memory, Liri – no coincidence! Anyone who knows how triangles are built can recognise such components – perhaps even the way to the place where they were built.”
Liri: “Three points on my map, joined up – and I already have the most stable shape of all. And a piece of memory of home as well.”
“Triangle: angle sum always 180°. Perimeter P=a+b+c. Area A=base×h÷2. Types by angles (acute/right/obtuse) and by sides (equilateral/isosceles/scalene).”
What you learned today
α+β+γ=180°  ·  A=base×h÷2
Perimeter: add up all the sides
🌟

Chapter 36 mastered!

Liri now knows the angle sum, perimeter and area of every triangle.

Erklärus’s notebook – page 36:
“Triangle: angle sum always 180°. Perimeter: all sides added. Area: base times height, divided by 2 (half a rectangle).”
← Back to overview 36 – The Triangle