Secondary 1 G3 Mathematics Tutor Bukit Timah | Composite Solids and Hidden Surface Areas

Did you know? When two cubes are glued together, their combined volume is the sum of their volumes—but their outside surface area is less than the sum of their individual surface areas. The two faces pressed against one another become hidden inside the new solid. You cannot paint them from the outside.

For a Secondary 1 G3 Mathematics learner in Bukit Timah, this is a valuable way to understand composite solids rather than memorise another formula. It builds on cubes, cuboids, cylinders, plane areas and careful diagram reading. The difficult part is often deciding which faces are exposed and which are internal, not multiplying the dimensions.

Quick answer: if two solids are joined along exactly one matching contact region, add their individual total surface areas and subtract twice the contact area. Add their volumes if their interiors do not overlap. If there are several contact regions, account for every hidden face exactly once.

Why do we subtract the contact area twice?

Imagine placing one closed cuboid against another. Before joining them, the first has an exposed face where contact will occur. The second also has an exposed matching face.

The combined shape has neither of those faces exposed at that contact. When we add the original surface areas, we have counted each hidden face once. There are two, so we must subtract twice the shared contact area.

For one full-face joint, the rule is:

Combined external surface area = surface area A + surface area B − 2 × contact area.

This is a counting principle, not a universal shortcut for every complicated solid. Overlapping volumes, multiple joints or internal cavities may require a different decomposition.

Start with one cube

A closed cube with side length 3 cm has six square faces. Each face has area 3 × 3 = 9 cm².

Total surface area = 6 × 9 = 54 cm².

Its volume is 3 × 3 × 3 = 27 cm³.

Notice the different units. Surface area counts squares covering the exterior; volume counts cubic units filling the interior. If the learner writes 54 cm³ or 27 cm², the formula and unit have become confused.

Worked example 1: joining two equal cubes

Take two cubes, each with side 3 cm, and join them along one entire 3 cm by 3 cm face.

Each cube originally has surface area 54 cm². The contact area is 9 cm².

Combined external surface area = 54 + 54 − 2(9) = 108 − 18 = 90 cm².

Volume = 27 + 27 = 54 cm³, because the interiors do not overlap.

Check by looking at the new shape: the two cubes form a cuboid measuring 6 cm by 3 cm by 3 cm. Surface area = 2(6 × 3 + 6 × 3 + 3 × 3) = 2(18 + 18 + 9) = 90 cm². Both methods agree.

What if two faces do not match completely?

Sometimes a small block rests on the middle of a larger block. Only the part actually touching becomes hidden, not the whole top of the larger block.

Suppose a 4 cm by 3 cm base of a smaller cuboid rests flat on a larger 8 cm by 6 cm top surface. The contact area is only 4 × 3 = 12 cm².

The exposed portion of the larger top face remains 8 × 6 − 4 × 3 = 48 − 12 = 36 cm².

If you subtract the whole 48 cm² top face, you would incorrectly remove an area that is still visible. Contact is defined by overlap of the two touching faces, not by the size of the larger surface.

Worked example 2: one cuboid stacked on another

A base cuboid measures 8 cm by 6 cm by 2 cm. A smaller cuboid measuring 4 cm by 3 cm by 2 cm sits entirely on the top, sharing its 4 cm by 3 cm bottom face with the base.

First calculate the base’s original surface area:

2(8 × 6 + 8 × 2 + 6 × 2) = 2(48 + 16 + 12) = 152 cm².

The upper cuboid’s original surface area is 2(4 × 3 + 4 × 2 + 3 × 2) = 2(12 + 8 + 6) = 52 cm².

Contact area = 4 × 3 = 12 cm². Combined external surface area = 152 + 52 − 2(12) = 180 cm².

The original surface areas add to 204 cm², but 24 cm² consists of two hidden contact faces. This is why the final result is 180 cm².

Check the same stacked solid by exposed faces

The base cuboid retains its bottom face of 48 cm², four side faces totalling 56 cm², and an uncovered top region of 48 − 12 = 36 cm².

The top cuboid contributes its top face of 12 cm² and four side faces totalling 28 cm².

Total exterior area = 48 + 56 + 36 + 12 + 28 = 180 cm².

This second method reveals each visible panel rather than using an overall subtraction shortcut. When the two answers agree, the reasoning is easier to trust.

If they differ, inspect which face was omitted or counted twice.

Volume for the same stacked solid

The base volume is 8 × 6 × 2 = 96 cm³.

The upper volume is 4 × 3 × 2 = 24 cm³.

Since the upper block rests on the lower block without their interiors overlapping, the combined volume is 96 + 24 = 120 cm³.

No surface contact subtraction is required for volume because the shared boundary has zero thickness. Surface area and volume respond differently to the same joined arrangement.

A useful diagnostic is to ask the child why subtracting 12 cm² from 120 cm³ makes no sense. The quantities have different dimensions and units.

Surface area depends on what the question wants painted

Suppose the stacked solid from the previous example is placed on a flat floor, and the question asks for the area of all exposed surfaces except the underside of the base.

The total geometrical exterior area is 180 cm². The bottom face measures 8 × 6 = 48 cm². Removing it from the painted area gives 180 − 48 = 132 cm².

The overall solid has not changed, but the requested measurement has. The underside is still part of the object’s total surface area; it is simply excluded from the painting instruction.

Always distinguish “total surface area” from “area to be painted” or “visible faces in the given arrangement.” A well-written question tells you which surfaces count.

Worked example 3: a small cube removed from a corner

Take a 4 cm cube. Remove a smaller 1 cm cube from one corner, cutting along three mutually perpendicular directions.

Originally, the large cube has surface area 6 × 4² = 96 cm².

The cut removes three exposed 1 cm by 1 cm patches, one from each of the three original outer faces meeting at the corner. That removes 3 cm² from the exterior.

It also reveals three new internal square faces inside the corner notch, each of area 1 cm². Those add 3 cm².

New surface area = 96 − 3 + 3 = 96 cm². It is unchanged, even though the solid now has a notch.

Volume does change: 4³ − 1³ = 64 − 1 = 63 cm³.

This is a striking reminder that removing material does not always reduce surface area. The location and shape of the cut determine what becomes exposed.

What if the cut is a notch in the middle of a face?

Removing a block from a face rather than a corner may expose more new internal faces than the face area removed.

Imagine a rectangular recess cut into the top of a solid, without going all the way through. The original opening area disappears from the flat top, but the base of the recess replaces that same area and its sidewalls add further area.

Under a simple rectangular recess model, the total exposed surface area therefore increases by the area of the recess sidewalls.

This is not the same as a corner cut, where removed exterior patches can be replaced by equal-area interior patches. A tutor should have students sketch the actual cut rather than assume all removals behave alike.

Worked example 4: a cylinder placed on a cuboid

Consider a cuboid 10 cm long, 8 cm wide and 4 cm high. A right circular cylinder of radius 2 cm and height 3 cm is fixed upright to its top face, entirely within that top face.

Cuboid surface area = 2(10 × 8 + 10 × 4 + 8 × 4) = 2(80 + 40 + 32) = 304 cm².

Cylinder total surface area before joining = 2πr² + 2πrh = 2π(2²) + 2π(2)(3) = 8π + 12π = 20π cm².

The circular contact area is π × 2² = 4π cm², counted once as part of the cuboid top and once as the cylinder bottom.

Combined external surface area = 304 + 20π − 2(4π) = 304 + 12π cm².

This is exact; a decimal approximation can be given when the question requests one.

A second check for the joined cylinder

Begin with the entire cuboid exterior, 304 cm². The cylinder covers a circular patch of 4π cm² on the cuboid’s top, so remove that patch.

The attached cylinder contributes its curved area 2πrh = 12π cm² and its exposed circular top of 4π cm². Its bottom is hidden.

So the new exterior area is 304 − 4π + 12π + 4π = 304 + 12π cm².

Both counting routes agree. Notice how the removal of one 4π patch and addition of the cylinder’s 4π top cancel numerically; the curved area is the net increase.

The combined volume is also straightforward: cuboid volume 10 × 8 × 4 = 320 cm³, cylinder volume π × 2² × 3 = 12π cm³, giving 320 + 12π cm³.

Exposed circular bases: one or two?

For a free-standing closed cylinder, count two circular bases and the curved side.

For a cylinder fixed by one entire base to another solid, that attached base is not exposed. Its top circular base usually remains exposed, assuming nothing else covers it.

For an open-top cylinder with a base, the top circular disk is not part of its material surface. Count its curved side and bottom, following the question’s model.

These are different physical or idealised objects. The cylinder formula is only useful when you know which surfaces the problem includes.

For foundations, see volume and surface area of prisms and cylinders.

Convert area and volume units correctly

One metre contains 100 centimetres. For area, both length dimensions scale: 1 m² = 100² cm² = 10,000 cm².

For volume, all three dimensions scale: 1 m³ = 100³ cm³ = 1,000,000 cm³.

If the combined solid has volume 120 cm³, its volume in cubic metres is 120/1,000,000 = 0.00012 m³.

You cannot convert the contact area of 12 cm² by the same factor as the volume. Write the square or cubic unit at each calculation stage.

A surprising number of otherwise correct mensuration solutions go wrong during a final unit conversion.

Five checks before calculating any composite solid

• Do the component solids actually overlap in volume, or do they merely meet along faces?
• Which faces or portions of faces are in contact and therefore hidden?
• Is the question asking for all outside surfaces, a painted subset or only curved area?
• Do all components use consistent units?
• Can a second route—counting exposed faces directly—confirm the subtraction method?

In a small-group lesson, ask one learner to calculate from the separate solids and another to list the exposed faces. Compare their totals and locate any disagreement.

The goal is not to memorise “subtract twice” mechanically. The goal is to understand exactly which two patches have become internal.

Ten-question mini diagnostic

• A. Find the surface area of a cube with side 3 cm.
• B. Find its volume.
• C. Join two such cubes along one full square face. Find the combined external area.
• D. Find the combined volume of those two cubes.
• E. A cuboid 8 × 6 × 2 cm has what surface area?
• F. A cuboid 4 × 3 × 2 cm has what surface area?
• G. Stack the smaller cuboid on the larger along a 4 × 3 cm face. Find the contact area.
• H. Find the external surface area of that stacked solid.
• I. Find its combined volume.
• J. If the underside of the larger cuboid is not painted, find the painted area of the stacked solid.

Answers with reasons

A. 6 × 3² = 54 cm².

B. 3³ = 27 cm³.

C. 54 + 54 − 2(9) = 90 cm².

D. 27 + 27 = 54 cm³.

E. 2(48 + 16 + 12) = 152 cm².

F. 2(12 + 8 + 6) = 52 cm².

G. 4 × 3 = 12 cm².

H. 152 + 52 − 24 = 180 cm².

I. 96 + 24 = 120 cm³.

J. 180 − 48 = 132 cm².

Ask the student why C subtracts 18 cm² but D subtracts nothing. Their answer reveals whether they can distinguish surface area from volume, rather than simply recall formulas.

A focused fifteen-minute tutorial

Start by drawing two separate cubes and shading one contact face on each. Ask the learner to count which faces will disappear when the cubes are joined.

Then draw the stacked-cuboid example. Have the child calculate the surface area by two routes: sum and subtract contact faces, and direct exposed-face counting.

Finish with a changed question excluding the underside or joining a cylinder. A different contact shape should not change the underlying reasoning.

At the next lesson, remove the worked template and ask the pupil to explain what gets counted before doing arithmetic.

What Bukit Timah parents should ask a tutor

Ask whether a student can locate and label each hidden face, work with partial-face contact, and distinguish a total-surface-area question from a painting-area question. These steps matter more than multiplying quickly.

The immutable eduKateSG Secondary 1 Mathematics small-group reference describes three-student tutorials near Sixth Avenue MRT that emphasise inspecting written working. The Bukit Timah G3 Mathematics tuition overview provides locality-specific context. Parents should verify the class fit and current schedule directly.

The MOE G2 and G3 Mathematics syllabus includes surface area and volume of prisms and cylinders and problems involving composite solids at Secondary 1 G3 level.

Frequently asked questions

### Why subtract twice the contact area?

Because each original solid includes one face of the contact in its surface-area total. Both faces become internal after joining.

### Do I also subtract contact area when finding volume?

No, if the interiors are nonoverlapping and the solids only share a boundary. Add their volumes. If the solids overlap in three-dimensional space, the overlapping volume must be handled separately.

### Can removing a small cube leave surface area unchanged?

Yes. Removing a small cube from a corner of a larger cube can remove and reveal equal total areas, although the volume decreases.

### Is the underside part of total surface area?

Yes. It counts unless the question specifically excludes it, such as when asking only for painted or visible surfaces.

### Why are area and volume units different?

Area measures a two-dimensional surface in square units; volume measures three-dimensional space in cubic units.

Keep learning through the series

• Nets of cubes, cuboids and prisms
• Volume and surface area of prisms and cylinders
• Perimeter and composite plane-figure areas
• Significant figures and estimation

The best first question for a composite solid is not “Which formula?” It is “Which faces can I touch from the outside?” Once those are clear, the surface-area calculation becomes easier to trust.