Composite solids · Code with Martin

Cut and add

A tower with a pointed roof. Ice cream in a cone. A pencil with a sharpened tip. The world is not made of clean cubes and spheres — it is made of composite solids: known shapes glued together.

And that is exactly how they are calculated. No new formulae today. Just one strategy: cut the solid in your head into pieces you already know — a cuboid, a cylinder, a pyramid, a cone, a hemisphere — calculate each separately and add.

For volume that holds literally: volume of the whole = the sum of the volumes of the parts. Air inside a tower = air in the cuboid + air in the pyramid roof.

For surface watch: adding everything would be a mistake. At the place where the parts glued, the area disappeared inside — a circle between a cylinder and a hemisphere nobody sees from outside. You only count what is really visible from outside.

This is the most practical lesson of solid geometry — and also a common format of an entrance-exam question worth more marks. Come and cut.

Jonáš helps dad paint a wooden fence post: a prism ended with a pyramid point. “How much paint shall we buy?” dad asks. Jonáš draws the post: “Four rectangular sides of the prism, four triangles of the point. Nothing at the bottom — the post sits in the ground. And the top base of the prism nothing either — the point sits on it.” Dad smiles: “So we do not paint surfaces that are not seen. That is exactly how you calculate surface: only what the brush can reach.”

💡

Always start with a sketch and a cut: with a line mark where one part ends and the other starts. Next to it write a list of parts. Only then reach for formulae.

A strategy for composite solids

Step 1: Recognise the parts. Look at the solid and name the pieces: “That is a cylinder and a hemisphere on it.” “That is a cuboid and a pyramid on it.” A sketch with a dividing line.

Step 2: List the sizes of each part. Watch shared sizes: the radius of the cylinder = the radius of the hemisphere on it. Height of the whole = height of the cuboid + height of the pyramid, so sometimes you have to find a part’s height by subtracting.

Step 3: Volume = a plain sum. V = V₁ + V₂. It always works. There is also subtracting: a pipe = a large cylinder minus a drilled small cylinder.

Step 4: Surface = only visible faces. Walk the solid in your head from outside like a painter: which faces will the brush touch? Leave out glued bases. For a tower (cuboid + pyramid): 4 sides of the cuboid + the bottom (if it is visible) + 4 triangles of the roof. The top base of the cuboid and the base of the pyramid disappeared in the glue.

Step 5: Units and a sense check. All parts in the same units! When the cylinder is in metres and the point in centimetres, unify first. And compare the result with an estimate: a tower cannot have a smaller volume than the cuboid it is made of.

Example 1: the volume of a tower (cuboid + pyramid)

Question: A little tower has the shape of a cuboid 4 × 4 × 9 m, on it sits a square pyramid 4 × 4 m with height 3 m. What is the volume of the tower?

Method: I cut: cuboid + pyramid.

Cuboid: V₁ = 4 · 4 · 9 = 144 m³.

Pyramid: V₂ = (4 · 4 · 3)/3 = 48/3 = 16 m³.

Sum: V = 144 + 16 = 160 m³.

Sense check: the roof added only 16 m³ to 144 — the point is a small part of the whole, it matches.

Answer: The volume of the tower is 160 m³.

Example 2: ice cream (cone + hemisphere)

Question: The cone is a cone with radius 3 cm and height 10 cm, on top sits a hemisphere of ice cream with the same radius. How many cm³ of ice cream is there in total (the cone is full)?

Method: Shared size: r = 3 cm for both parts.

Cone: V₁ = (3.14 · 9 · 10)/3 = 282.6/3 = 94.2 cm³.

Hemisphere: a whole sphere (4/3) · 3.14 · 27 = 113.04 cm³, half V₂ = 56.52 cm³.

Sum: V = 94.2 + 56.52 = 150.72 cm³.

Answer: There is about 150.7 cm³ of ice cream — more than a third of that is the scoop on top.

Example 3: the surface of a post (visible faces)

Question: A post: a cuboid 20 × 20 × 100 cm, on top a pyramid point with base 20 × 20 cm and face height 15 cm. The post stands on the ground. How many cm² do you paint?

Method: A painter’s inspection: 4 sides of the cuboid — yes. Bottom — no (it stands on the ground). Top base of the cuboid — no (the point sits on it). 4 triangles of the point — yes.

Sides: 4 · (20 · 100) = 8000 cm².

Triangles: 4 · (20 · 15)/2 = 4 · 150 = 600 cm².

Sum: 8600 cm².

Answer: You paint 8600 cm², so 0.86 m². Glued and covered faces are not counted.

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For surface do not add glued faces. The sum of the surfaces of both parts is almost always wrong — the glue place is hidden inside. Walk the solid with a painter’s eyes: what the brush does not paint does not belong in the surface. For volume, on the other hand, you add everything — the air inside is one big shared home.

Practice: now you do it 🎲

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

Practise the volume of a cylinder — it is the most common part of composite solids. Calculate with π ≈ 3.14, write with a comma.

Practise the questions

On paper

A sketch with a dividing line and a list of parts for every problem. Then the formulae.

A little house: a cuboid 6 × 4 × 3 m with a pitched roof in the shape of a triangular prism (triangle: base 4 m, height 1.5 m, length 6 m — volume = area of the triangle times length). Calculate the volume of the whole house.
A silo: a cylinder with radius 2 m and height 5 m, on top a hemisphere with the same radius. Calculate the total volume.
A pencil: a cylinder with radius 0.4 cm and length 15 cm, the tip a cone with the same radius and height 1.5 cm. Volume of the whole pencil? Round to one decimal place.
For the silo from problem 2 list the visible faces (it stands on the ground). Which faces do not count in the surface and why?

Now you 💪

Every composite solid you first cut with a sketch into known parts.
You add all volumes, for surface only visible faces.
You watch shared sizes and the same units across parts.

✅

Done when: You have calculated the volumes of three composite solids and for one correctly sorted the visible faces of the surface.

What to take from this lesson

A composite solid = known parts glued together. You do not need new formulae.
The volume of the whole is the sum of the volumes of the parts (sometimes a difference — holes).
Surface = only faces visible from outside. The glue disappeared inside.
A sketch with a dividing line and a list of parts is half the solution.
Shared sizes (a common radius, a split height) read from the question carefully.