A solid with an apex

On a passing asteroid a striking mountain rose up – a wide, square base whose four sides came together to a single peak. “I know a structure like that! At home there was a landing module with exactly this shape.”

Erklärus: “That is a pyramid. Let’s find out step by step how to work it out.”

🔺 The pyramid with a square base

A pyramid has a base (here a square with side a) and four triangular side faces that meet at an apex.

h Apex a
Formula – base area (square)
B = a × a = a²

📐 The net – a square with four triangles

Liri: “What does the pyramid look like unfolded?”
Erklärus: “A square as the base, surrounded by four isosceles triangles – one for each side face. Each triangle has the base side a and the slant height h_a.”
Formula – lateral surface area
L = 4 × (a × ha) ÷ 2 = 2 × a × ha

📦 Surface area and volume

Unlike with the prism, the base counts only once here – the pyramid has no top face, only the single apex.

Formula – surface area
SA = B + L
Liri: “And the volume? Surely not simply B times h like with the prism?”
Erklärus: “Well spotted – a pyramid fills only a third of the prism with the same base and height.”
Formula – volume
V = (1/3) × B × h
a=6 cm, h=9 cm – volume?
🌌 Pyramids in the universe

Real space probes also sometimes use pyramid-shaped landing legs or heat shields – the shape spreads forces evenly over a wide base on landing.

❌ Common mistake

The third in the volume is often forgotten! V is not B×h as with the prism, but only a third of that.

📝 Exercise 1
a=4 cm – how big is the base area B?
📝 Exercise 2
B=25 cm², h=6 cm – volume?
📝 Exercise 3
a=5 cm, h_a=8 cm – lateral surface area L?

The landing module worked out

Liri wrote the measurements of the landing module onto her star map.

Liri: “A third of the prism – I would never have thought that! But now it makes sense: the pyramid comes to a point.”
Erklärus: “Exactly. And your map is now richer by another detail.”
“Pyramid: B=a². L=2×a×h_a. SA=B+L. V=(1/3)×B×h.”
What you learned today
V = ⅓ × B × h
A third of the prism with the same base and height
🌟

Chapter 53 mastered!

Liri can now work out every pyramid.

Erklärus’s notebook – page 53:
“Pyramid: B=a², V=⅓Bh. Only one base counts in SA=B+L.”
← Back to overview 53 – The Pyramid