Prisms: surface area and volume

Pull a shape up into a height and you have a solid

So far we have calculated flat shapes. Today we give them a third dimension. A prism is born when you pull a plane shape — the base — up perpendicular into a height. A triangular base gives a triangular prism (the shape of a tent or a Toblerone), a quadrilateral one a cuboid, a hexagonal one a pencil.

Two questions that keep coming back with solids:

How much fits in it? That is volume V. It calculates beautifully: V = area of the base × height. A layer of the base 1 cm high, and there are as many such layers as the height measures.

How much material is on its walls? That is surface area S: S = 2 × base + lateral surface. The lateral surface is the side walls — when you unfold a prism, the lateral surface is one rectangle as wide as the perimeter of the base and as high as the prism.

See why we trained areas? The base is a triangle, a rectangle or a trapezium — and you already know all of those.

Ema and Jonáš make stands in the shape of a triangular prism for a fair. Base: a right-angled triangle with legs 6 and 8 cm, length of the prism 12 cm. Jonáš cuts and Ema counts cardboard: “Two triangular bases: 2 × 24 = 48 cm². Lateral surface: the perimeter of the base is 6 + 8 + 10 = 24 cm, times length 12… 288 cm². In total 336 cm² per piece.” Jonáš raises an eyebrow: “And will the unfolded one fit in a 20 × 30 cm box?” Ema draws a net of the solid on paper. “It will. But only if we split the lateral surface.”

Volume and surface area step by step

Volume: V = Sp × v (area of the base times the height of the solid).

Step 1: decide what the base is. On a prism those are the two identical parallel faces — a prism in a picture can happily “lie on its side” and have the bases at the sides. Step 2: calculate the area of the base with the formula for its shape. Step 3: multiply by the height of the prism (the distance of the bases).

Units: cm² times cm = cm³, cubic centimetres. Into 1 cm³ fits 1 ml of water — volumes and litres are relatives: 1 litre = 1 000 cm³ = 1 dm³.

Surface area: S = 2 × Sp + lateral surface.

Step 1: area of the base (you already have it from the volume). Step 2: lateral surface = perimeter of the base × height of the prism. The unfolded lateral surface is a rectangle — that is why. Step 3: add: two bases + lateral surface. Units cm².

A cuboid as a special case: base a rectangle a × b, height c. Volume V = a × b × c, surface area S = 2 × (ab + bc + ac). Formulas you may know — now you know where they come from.

A closing observation: volume grows with the third power — a box with double edges holds 8× more. That is why a big pack is often better on material.

A cuboid aquarium 40 × 25 × 30 cm: volume in litres

Step 1: the base is a rectangle 40 × 25 cm: Sp = 1 000 cm².

Step 2: volume: V = 1 000 × 30 = 30 000 cm³.

Step 3: convert to litres: 1 000 cm³ = 1 litre, so 30 000 cm³ = 30 litres.

Result: 30 litres. Check with an estimate: a bucket holds 10 litres — the aquarium looks like three buckets, that checks out. Converting cm³ → litres is crossing out three zeros; you will need it with every tank and pool.

A tent — a triangular prism: base with base 2 m and height 1.5 m, length 3 m

The base is the triangular gable of the tent, the “height of the prism” is its length.

Step 1: area of the base: Sp = (2 × 1.5) : 2 = 1.5 m².

Step 2: volume: V = 1.5 × 3 = 4.5 m³.

Result: 4.5 m³ of air. A prism can happily lie — the bases are at the front and back, not at the bottom. The spotting sign of the bases: two identical faces, between which the distance is the same everywhere.

Surface area of a triangular prism from the scene: legs 6 and 8, hypotenuse 10, length 12 cm

Step 1: the base — a right-angled triangle: Sp = (6 × 8) : 2 = 24 cm².

Step 2: perimeter of the base: 6 + 8 + 10 = 24 cm.

Step 3: lateral surface: 24 × 12 = 288 cm².

Step 4: surface area: S = 2 × 24 + 288 = 48 + 288 = 336 cm².

Result: 336 cm². Notice the split of work: the perimeter of the base feeds the lateral surface, the area of the base feeds the volume (V = 24 × 12 = 288 cm³). Who mixes perimeter and area mixes both answers.

Practice: now you do it 🎲

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

Revise the area of a rectangular base — the volume of a cuboid sits on it. Write only the number in cm².

Practise the questions

On paper: boxes and tents

For each question first write what the base is and what shape it has. Only then the formulas.

1. A cuboid 15 × 10 × 8 cm: calculate volume and surface area. Watch the units (cm³ vs. cm²).

2. A triangular prism: the base is a triangle with base 10 cm and height 6 cm, the height of the prism 20 cm. Volume?

3. A paddling pool in the shape of a cuboid 3 m × 2 m × 0.5 m: how many litres of water does it hold? (1 m³ = 1 000 litres.)

4. Draw a net of a cuboid 6 × 4 × 3 cm, label the sizes of all six faces and calculate the surface area by adding the areas of the net. Check with the formula.

Now you 💪

1. I know V = area of the base × height and I know why it holds (stacking the base).
2. I build surface area from two bases and the lateral surface (perimeter of the base × height).
3. I convert cm³ to litres and m³ to litres.

What to take from this lesson

• A prism = a base pulled up perpendicular into a height.
• Volume: V = Sp × v, cubic units (cm³).
• Surface area: S = 2 bases + lateral surface; lateral surface = perimeter of the base × height.
• A litre = 1 000 cm³ = 1 dm³.
• Units tell you whether you are calculating volume or surface area.

© 2026 Ing. Martin Polak / AlgoRhino · 内容使用条款