Solve surface-area and volume problems with combined solids

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Break a three-dimensional shape into familiar parts, then count only what is needed.

A storage box with a raised section, or a block made by joining two smaller blocks, is a combined solid. It can be easier to measure the whole shape by splitting it into solids you already know. The key is to keep track of what is inside the combined shape and what remains on the outside. This lesson reviews rectangular prisms and shows how to use them to find volume and surface area.

What you will learn

Identify familiar solids that make up a combined solid.
Find the volume of a combined solid by adding or subtracting the volumes of its parts.
Find the surface area by counting only the outside faces.
Use correct square units for surface area and cubic units for volume.

1. Grade 9 bridge: recognize and measure a solid

A solid is a three-dimensional object. A rectangular prism has six rectangular faces, like a box. Its length, width, and height describe the distances along its three directions.

Volume measures how much space is inside a solid. For a rectangular prism, multiply its length by its width and height. Surface area measures the total area covering the outside. Add the areas of all the outside faces.

Area is measured in square units, such as square centimetres. Volume is measured in cubic units, such as cubic centimetres. The units help you tell which quantity you have found.

V=lwh

Volume describes space inside a solid.
Surface area describes the area of its outside covering.
Use cubic units for volume and square units for surface area.

2. Split the solid and decide what counts

A combined solid is made by joining two or more familiar solids. First, sketch or picture the shape and identify its parts. Mark the dimensions you know. If a dimension is missing, use the diagram and the way the parts meet to work it out.

For volume, find the volume of each part and add the amounts when the parts fit together without overlapping. If a piece has been cut out, subtract the missing volume instead. Do not count a shared face as a loss of volume: the joined parts still fill both regions.

For surface area, imagine wrapping the entire outside of the finished shape. A face where two solids touch is hidden inside, so it is not part of the outside surface. If you calculate each solid’s full surface area separately, subtract the area of each hidden face from both solids. That means a shared contact area is subtracted twice.

A useful way to organize the work is to list each part’s volume and its exposed faces. A face is exposed when it is visible on the outside of the finished shape. A face is hidden when another part covers it.

V_{\text{combined}}=V_1+V_2

Add volumes for joined parts that do not overlap.
Do not include faces that touch inside the combined solid.
When adding full surface areas, subtract both copies of each shared face.

3. A visual plan and the main equations

The table gives a plan for a combined solid made from two rectangular prisms. The shared face is one rectangle on each prism. Those two rectangles meet and become hidden when the prisms are joined.

For a rectangular prism with length

l

, width

w

, and height

h

, the volume is the product of its three dimensions. Its surface area comes from adding the areas of three pairs of equal rectangular faces: two

lw

faces, two

lh

faces, and two

wh

faces.

The symbol

A

means area. The symbols

V

and

S

can represent volume and surface area. Use the same units for all lengths before calculating. For example, convert metres to centimetres if the dimensions are given in both units.

S=2(lw+lh+wh)

A rectangular prism has three pairs of equal faces.
Check that all measurements use the same length unit.
Write the unit with the final answer.

4. Guided example and independent practice

Use the worked example to see how volume and surface area need different counting. The volume includes all space occupied by both blocks. The surface area includes only faces that are still exposed. Afterward, try the practice prompts without looking at the worked steps.

For volume, include each block’s interior space.
For surface area, remove the hidden contact faces.

Plan for two joined rectangular prisms

Quantity	What to calculate	What to do with the shared face
Volume	Find each prism’s length × width × height	Add the volumes if the parts do not overlap
Surface area	Find each prism’s outside face areas	Subtract both hidden copies of the contact area

Worked example

A raised block on a base

A rectangular prism measuring

8\text{ cm}

by

5\text{ cm}

by

3\text{ cm}

has a smaller rectangular prism measuring

4\text{ cm}

by

5\text{ cm}

by

2\text{ cm}

placed on top. The smaller prism rests fully on the base. Find the volume and outside surface area of the combined solid.

Find the volumes
Calculate each prism’s volume by multiplying its three dimensions. The blocks do not overlap, so add their volumes.
$V_1=8(5)(3)=120\text{ cm}^3,\quad V_2=4(5)(2)=40\text{ cm}^3,\quad V=120+40=160\text{ cm}^3$
Find each full surface area
Use the rectangular-prism surface-area equation for each block. These totals include every face, including the faces that will be hidden where the blocks touch.
$S_1=2(8\cdot5+8\cdot3+5\cdot3)=158\text{ cm}^2,\quad S_2=2(4\cdot5+4\cdot2+5\cdot2)=76\text{ cm}^2$
Remove the hidden faces
The contact rectangle measures $4\text{ cm}$ by $5\text{ cm}$ , so its area is $20\text{ cm}^2$ . It is counted once on the base and once on the top block. Subtract both copies.
$S=158+76-2(4\cdot5)=194\text{ cm}^2$

Answer: The combined solid has a volume of

160\text{ cm}^3

and an outside surface area of

194\text{ cm}^2

.

Check: The surface-area total is less than the sum of the two separate surface areas because the

20\text{ cm}^2

contact face on each block is hidden. The volume is the sum of the two non-overlapping blocks.

Common mistakes and how to avoid them

Adding the full surface areas without removing the contact faces.

Correction: Faces touching inside the combined solid are not outside faces. Subtract the area of each hidden copy.

Subtracting the shared face only once after adding both full surface areas.

Correction: The contact area appears once in each solid’s surface-area total, so subtract it twice.

Using square units for volume or cubic units for surface area.

Correction: Surface area uses square units because it measures faces. Volume uses cubic units because it measures space.

Adding volumes when the parts overlap.

Correction: Adding works only when the parts occupy separate regions. If a piece overlaps or is removed, account for that region so it is not counted twice or included when it is missing.

Lesson summary

Break a combined solid into familiar parts, such as rectangular prisms.
Find the volume of each part and add the volumes when the parts do not overlap.
For surface area, include only faces on the outside of the finished solid.
If you add full surface areas, subtract each shared contact area twice.
Check dimensions and use square units for surface area and cubic units for volume.

Check your understanding

Question 1

Two non-overlapping rectangular prisms are joined. What should you do to find their combined volume?

Add the volume of each prism.
Subtract the smaller volume from the larger volume.
Add their surface areas and divide by two.
correctIndex": 0, "explanation": "The two prisms occupy separate regions, so their volumes combine by addition."}

Show answer and explanation

Add the volume of each prism.

The two prisms occupy separate regions, so their volumes combine by addition.

Question 2

Two blocks touch along a rectangular face. What happens to that contact face when they are joined?

It is counted twice in the outside surface area.
It is hidden and is not part of the outside surface area.
It becomes part of the volume’s surface area.
correctIndex": 1, "explanation": "The touching face lies inside the combined solid, so it is not exposed on the outside."}

Show answer and explanation

It is hidden and is not part of the outside surface area.

The touching face lies inside the combined solid, so it is not exposed on the outside.

Question 3

A prism has dimensions

3\text{ cm}

,

2\text{ cm}

, and

4\text{ cm}

. What is its volume?

$9\text{ cm}^3$
$24\text{ cm}^3$
$24\text{ cm}^2$
correctIndex": 1, "explanation": "Multiply the three dimensions: $3(2)(4)=24$ . The answer is a volume, so its unit is cubic centimetres."}

Show answer and explanation

24\text{ cm}^3

Multiply the three dimensions:

3(2)(4)=24

. The answer is a volume, so its unit is cubic centimetres.

Key terms

Combined solid: A three-dimensional shape made by joining familiar solids.
Exposed face: A face that is visible on the outside of the finished solid.
Hidden face: A face covered by another part when solids are joined.
Volume: The amount of space inside a solid.
Surface area: The total area of the outside faces of a solid.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M11. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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