A pyramid: surface area and volume

A pyramid on the desk

You know a cube and a cuboid, a cylinder too. Today a solid the whole world knows from Egypt is added: a pyramid. A base at the bottom, a point at the top, faces made of triangles.

We will focus on the most common type: a square pyramid. Exactly like the pyramids at Giza.

The volume of a pyramid has a beautifully simple formula: V = (Sp · v) / 3. Sp is the area of the base, v is the height — the perpendicular distance from the base to the point. That dividing by three is not a memory trick: into a prism with the same base and height, exactly three pyramids fit. A pyramid is a third of its prism.

The surface area is built: a square base + four triangular faces. S = Sp + Spl, where Spl is the lateral surface (those four triangles).

The only trap: a pyramid has two different heights — the height of the solid (inside, to the point) and the height of a face (along the surface of a triangle). Volume takes the first, surface the second. Today you will learn not to mix them up.

Ema is folding a gift box in the shape of a pyramid from paper. “I need to know how much paper — and how many sweets fit inside.” Adam unfolds the net: a square and four triangles. “Paper is surface: the square plus four triangles. Sweets are volume: area of the base times height, divided by three.” Ema raises an eyebrow: “Divided by three? So that little fits inside?” Adam nods: “The point at the top eats space. A pyramid is only a third of the box it grows from.”

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Before you calculate, decide what the question asks: how much FITS (volume, units cm³) or how much MATERIAL (surface, units cm²). The units of the result will tell you at once if you picked the right formula.

Volume and surface step by step

Volume of a square pyramid:

Step 1: Area of the base. A square with side a: Sp = a². Step 2: Multiply by the height of the solid v (perpendicular from the centre of the base to the point). Step 3: Divide by three. V = (a² · v) / 3.

Example: a = 6 cm, v = 5 cm. Sp = 36 cm². V = (36 · 5)/3 = 180/3 = 60 cm³.

Surface:

Step 1: Base: Sp = a². Step 2: One triangular face: base a, height of the face va (measured along the face from the base edge to the point). Area = (a · va)/2. Step 3: There are four faces: Spl = 4 · (a · va)/2 = 2 · a · va. Step 4: Add: S = a² + 2 · a · va.

Example: a = 6 cm, va = 5 cm. S = 36 + 2 · 6 · 5 = 36 + 60 = 96 cm².

Those two heights: the height of the solid v goes inside, the height of the face va along the surface — it is always longer than v (Pythagoras joins them through half a side of the base). The question usually gives both, but you have to sort them right: v into volume, va into surface.

Example 1: the volume of a tent

Question: A tent has the shape of a square pyramid with side 3 m and height 2 m. How much air is inside?

Method: It asks for volume (how much fits), the units will be m³.

Base: Sp = 3² = 9 m².

Volume: V = (9 · 2)/3 = 18/3 = 6 m³.

Sense check: a cuboid 3 × 3 × 2 m would have 18 m³; the tent is a third of it, 6 m³. It matches.

Answer: There is 6 m³ of air in the tent.

Example 2: the surface of a gift box

Question: A paper pyramid has a square base with side 10 cm and a face height of 8 cm. How much paper is needed for the whole box including the bottom?

Method: It asks for material → surface, units cm².

Base: 10² = 100 cm².

One face: (10 · 8)/2 = 40 cm². Four faces: 160 cm².

Total: S = 100 + 160 = 260 cm².

Answer: 260 cm² of paper is needed. Notice: the question had the face height (8 cm along the surface) — that is exactly the one that belongs in surface.

Example 3: both heights in one question

Question: A pyramid has a square base with side 6 cm, solid height 4 cm and face height 5 cm. Calculate the volume and the surface.

Method: I sort the heights: 4 cm goes inside → volume. 5 cm along the face → surface.

Volume: Sp = 36 cm², V = (36 · 4)/3 = 48 cm³.

Surface: a face (6 · 5)/2 = 15 cm², four faces 60 cm², plus the base 36 cm². S = 96 cm².

Check: the face height (5) is bigger than the solid height (4) — that is how it should be, along a slanted face it is further to the point.

Answer: V = 48 cm³, S = 96 cm².

⚠️

Do not mix up the height of the solid and the height of a face. Volume takes the height of the solid (perpendicular inside to the point), surface takes the height of a face (along the slanted triangle). If you swap them, both results are wrong. A helper: the face height is always longer — a slanted path is longer than a perpendicular one.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Volumes of square pyramids: a² times v, divided by three. Write a whole number in cm³.

Practise the questions

On paper

A sketch of a pyramid for every problem. Mark which height is which.

A pyramid: base 9 × 9 cm, solid height 10 cm. Calculate the volume and check that it is a third of a cuboid 9 × 9 × 10.
A pyramid: base 8 × 8 cm, face height 6 cm. Calculate the surface including the base.
A pyramid: base 6 × 6 cm, solid height 4 cm, face height 5 cm. Calculate the volume and the surface and check that you did not mix up the heights.
A pyramid tent: base 2.5 × 2.5 m, height 1.8 m. How many m³ of air is inside? Round to one decimal place.

Now you 💪

You can explain why the volume of a pyramid is divided by three.
You tell apart the height of the solid (volume) and the height of a face (surface).
You watch the units: volume in cm³ or m³, surface in cm² or m².

✅

Done when: You have calculated the volume and the surface of a pyramid, the heights correctly sorted and one check through a third of a cuboid.

What to take from this lesson

Volume of a pyramid: V = (area of the base · height) / 3. A pyramid is a third of a prism.
Surface: the base + four triangular faces.
The height of the solid goes inside (volume), the height of a face along the surface (surface).
The face height is always longer than the solid height.
Units give away the formula: cm³ = volume, cm² = surface.