A slanting shadow

The Sun cast the shadow of a rectangular solar panel at a slant onto the ground – the rectangle became a lopsided shape whose opposite sides, however, still stayed exactly parallel to each other. “I know that! At home, struts were sometimes bent into exactly this shape when the wind pushed them sideways.”

Erklärus: “That is called a parallelogram. Let’s find out what rules apply to it.”

▱ What is a parallelogram?

A parallelogram is a four-sided shape in which the opposite sides are parallel and equal in length.

h (height) base A(0,0) B(5,0) C(7,3) D(2,3)
Liri: “So like a rectangle that has been pushed sideways?”
Erklärus: “Exactly that picture! The sides stay the same length and parallel – only the angles change. And no matter how slanted: the height h always stands at right angles to the base.”

📐 Properties of the angles

Erklärus: “Opposite angles are always the same size. And two neighbouring angles always add up to 180°.”
α β β α
Angle rules

α opposite α, β opposite β  ·  α + β = 180°

📏 Perimeter and area

The perimeter is the sum of all four sides – and since opposite sides are the same length:

Formula – perimeter
P = 2 × a + 2 × b
Liri: “And the area? Surely not simply a times b, because it’s slanted?”
Erklärus: “Good objection! Cut off the triangle on one side and put it back on the other side – then the parallelogram becomes a rectangle with the same area.”
cut off… g h

The orange triangle moves from right to left – the area stays exactly the same.

Formula – area
A = base × h

h always stands at a right angle (90°) to the base.

base = 8 cm, h = 5 cm – what is the area?

🧮 Liri’s parallelogram calculator – interactive

Enter the two sides and the height – the perimeter and area appear straight away:

▱ Calculate a parallelogram
🌌 Parallelograms in the universe

When a rectangular solar sail is seen at an angle to the line of sight, it looks like a parallelogram in perspective – just like an obliquely lit shadow.

❌ Common mistake

The area is not side times side (a×b)! You need the height h (perpendicular distance), not the slanted side b.

📝 Exercise 1
a = 6 cm, b = 4 cm – what is the perimeter?
📝 Exercise 2
base = 10 cm, h = 3 cm – what is the area?
📝 Exercise 3
One angle in a parallelogram is 70°. How big is the neighbouring angle?

A shadow, understood

Liri now wrote the shape of the slanting shadow onto her star map with exact measurements.

Liri: “Even a shape that has been pushed sideways follows clear rules – the area stays base times height.”
“Parallelogram: opposite sides parallel & equal in length. Opposite angles equal, neighbouring angles=180°. P=2a+2b. A=base×h.”
What you learned today
A = base × h
Base times height – the height always stands at right angles to the base
🌟

Chapter 38 mastered!

Liri can now work out every parallelogram.

Erklärus’s notebook – page 38:
“Parallelogram: opposite sides & angles equal. A=base×h, P=2a+2b.”
← Back to overview 38 – Parallelogram