A rainbow made of ice dust

Tiny ice crystals drifted past the window and broke the starlight into a shimmering band of colour. “I know an effect like that!” cried Liri. “At home there was a sparkling crystal that spread out the light in exactly the same way – shaped exactly the same at the top and at the bottom, with smooth, rectangular sides.”

Erklärus: “That sounds like a prism. Let’s find out how to work out its shape.”

🔷 What is a prism?

A prism has two identical, parallel faces – the base B and the top face – joined by nothing but rectangles as side faces. The name comes from the shape of the base.

h Top face Base B
Key point

The base and the top face are identical and parallel. The side faces are rectangles. The name of the prism comes from the base – e.g. “triangular prism” when the base is a triangle.

📐 The net – the lateral surface as a rectangle

Liri: “If I unfold the prism, what do I get?”
Erklärus: “All the side rectangles together make one big rectangle – the lateral surface. Its width is the perimeter of the base p_B, its height is h.”
Formula – lateral surface area
L = pB × h

📦 Surface area and volume

The surface area is the lateral surface plus the base twice (once at the top, once at the bottom):

Formula – surface area
SA = 2 × B + L
Liri: “And the volume? Like with the cuboid – base times height?”
Erklärus: “Exactly, the principle stays the same, no matter what shape the base has.”
Formula – volume
V = B × h
Triangular prism: B=12 cm², h=5 cm – volume?
🌌 Prisms in the universe

Some satellite components and optical instruments in telescopes use real glass prisms to bend light and split it into its colours – the same geometric principle, only with light instead of volume.

❌ Common mistake

With the surface area the second base is often forgotten! Remember: twice B – once at the top, once at the bottom.

📝 Exercise 1
Base B=8 cm², height h=6 cm – volume?
📝 Exercise 2
Perimeter of the base p_B=10 cm, height h=7 cm – lateral surface area?
📝 Exercise 3
B=15 cm², L=40 cm² – what is the surface area (SA=2B+L)?

The crystal measured

Liri wrote the exact measurements of the crystal onto her star map – base, height, volume.

Liri: “No matter what shape the base has – the principle base times height always works!”
Erklärus: “That is exactly what makes prisms so versatile – and your memory so valuable.”
“Prism: base and top face identical & parallel. L=p_B×h. SA=2B+L. V=B×h.”
What you learned today
V = B × h
Base times height – for any shape of base
🌟

Chapter 51 mastered!

Liri can now work out every prism.

Erklärus’s notebook – page 51:
“Prism: base and top face identical. V=B×h applies to every shape of base.”
← Back to overview 51 – Prisms