E. Spatial Sense · Spatial Sense: Angles, Tessellations, Transformations, Scale, Pythagoras and Composite Measurement

Surface Area and Volume of Composite Objects

E2.3 A1About 50 minutes

Learning goals

I can break a composite 3D object into prisms and cylinders.
I can find the volume of a composite object by adding or subtracting volumes.
I can find the surface area of a composite object, leaving out the faces that are hidden where the parts join.
I can solve real problems about building, painting and filling composite objects.

Words to know

composite object
A 3D object made of two or more simpler objects, like a box with a triangular roof.

volume
The amount of space an object takes up, in cubic units (cm³, m³). For any prism or cylinder, V = area of base × height.

surface area
The total area of all the outside surfaces of an object, in square units.

hidden face
A face, or part of a face, that is covered where two parts of a composite object join. It is not part of the surface area.

capacity
How much a container can hold. 1 cm³ holds 1 mL, and 1 000 cm³ holds 1 L.

Remember from Grade 7

In Grade 7 you learned that the volume of any prism or cylinder is V = (area of the base) × height, you found the surface area of cylinders from their nets (SA = 2πr² + 2πrh), and you learned that 1 cm³ = 1 mL. In Grade 6 you found the surface area of prisms and pyramids. In Grade 8 you'll put these objects together and take pieces away.

Let's start with a doghouse

A family is building a doghouse: a box with a triangular roof. They want to know how much space is inside (volume) and how much outside surface they'll need to paint (surface area). The doghouse is a composite object: a rectangular prism with a triangular prism on top.

The doghouse: a rectangular prism (80 cm wide, 100 cm deep, 60 cm high) with a triangular prism roof on top. The triangle at the front has a base of 80 cm and a height of 30 cm, so each sloping side is 50 cm. Not to scale: the depth is drawn shorter and slanted to show the object in 3D.

Volume of a composite object

Add the volumes of the parts: V = V₁ + V₂.
If a part is removed (a hole, a hollow pipe), subtract its volume.
For an object that is the same shape all the way through (like an L-shaped step), you can find the area of the front face and multiply by the length.

Surface area of a composite object

Add the areas of the faces you can touch from the outside.
Leave out hidden faces where the parts are joined.
A useful trick: when a smaller object sits on top of a larger one, the top of the small object replaces the area it covers, so the visible top area doesn't change. You only add the small object's sides.

Two 1 cm cubes have 12 faces showing when they are apart. Glue them together and two faces get hidden (one on each cube), so the new object has 10 cm² of surface area. Volume, though, just adds: 1 + 1 = 2 cm³.

Worked example

Example 1: The doghouse

Find the volume of the doghouse, and the area to paint on the outside (everything except the floor).

Volume of the box: 80 × 100 × 60 = 480 000 cm³.
Volume of the roof: triangle area 80 × 30 ÷ 2 = 1 200 cm², times the length 100 cm = 120 000 cm³.
Total volume: 480 000 + 120 000 = 600 000 cm³ = 0.6 m³.
Paint: front and back walls are each a rectangle plus a triangle: 80 × 60 + 1 200 = 6 000 cm², so 2 × 6 000 = 12 000 cm².
Two side walls: 2 × (100 × 60) = 12 000 cm². Two roof panels: 2 × (50 × 100) = 10 000 cm².
The top of the box and the bottom of the roof are hidden, and the floor isn't painted. Total: 12 000 + 12 000 + 10 000 = 34 000 cm².

Volume 600 000 cm³ (0.6 m³); paint area 34 000 cm² (3.4 m²).

Worked example

Example 2: A concrete step

Two concrete steps are 90 cm wide. Seen from the side, they make an L-shape: a bottom block 60 cm long and 20 cm high, and a top block 30 cm long and 20 cm high sitting on the back half. How much concrete is needed?

Side face area: 60 × 20 + 30 × 20 = 1 200 + 600 = 1 800 cm².
The step is the same shape all the way across, so V = 1 800 × 90 = 162 000 cm³.
1 m³ = 1 000 000 cm³, so this is 0.162 m³.

162 000 cm³, or about 0.16 m³ of concrete.

Worked example

Example 3: A hollow pipe

A pipe is 200 cm long. Its outer radius is 5 cm and its inner radius is 4 cm. What volume of material is it made of?

Outer cylinder: 3.14 × 5 × 5 × 200 = 15 700 cm³.
Hole: 3.14 × 4 × 4 × 200 = 10 048 cm³.
Material: 15 700 − 10 048 = 5 652 cm³. (Shortcut: 3.14 × (25 − 16) × 200 = 3.14 × 9 × 200 = 5 652.)

5 652 cm³ of material.

Worked example

Example 4: A cylinder on a cube

A wooden stamp is a 10 cm cube with a cylinder handle (radius 3 cm, height 5 cm) glued to the middle of its top. Find its surface area.

Cube: 6 × 10 × 10 = 600 cm².
The handle covers a circle on the cube's top, but the handle's own top circle is exactly the same size. So the top area that shows is still 100 cm². Just add the handle's curved side.
Curved side: 2 × 3.14 × 3 × 5 = 94.2 cm².
Total: 600 + 94.2 = 694.2 cm².

694.2 cm²

Think like a mathematician

Sketch the parts separately and label their dimensions before you calculate.
Volume: add parts, subtract holes. Surface area: count only the outside faces.
Gluing two objects together always hides some area, so the surface area of the composite is less than the two surface areas added together. Volume, though, just adds.
Units: cm³ for volume, cm² for area. 1 000 cm³ = 1 L and 1 m³ = 1 000 000 cm³.

Hands-on activity

Build and measure a composite object

You need: two or three boxes or containers of different sizes (cereal box, tissue box, a can), tape, a ruler, a calculator. Optional: a measuring cup and dry rice.

Tape the objects together to make a composite object: for example, a tissue box on top of a cereal box, or a can standing on a box.
Measure every part. Sketch each part separately with its dimensions.
Calculate the total volume.
Calculate the surface area of the composite (leave out the hidden parts). Compare it with the sum of the separate surface areas. How much area is hidden?
Optional: fill the can with rice, then pour it into a measuring cup to check its capacity in millilitres. Compare it with your calculated volume (1 cm³ = 1 mL).

What to look for: volumes add exactly. The composite surface area is less than the separate surface areas added together; the difference is twice the area of contact (the hidden part on each object). The rice check should be close, but a little less because of the can's walls and the gaps between grains.

Just for fun

A pizza has radius z and thickness a. What is its volume?

Pi × z × z × a. (Check: that's π × r × r × h for a very flat cylinder!)

Your turn

A 2 cm cube sits on top of a box that is 4 cm × 3 cm × 2 cm. Find the total volume.

A roof is a triangular prism. The triangle has a base of 6 m and a height of 2 m, and the roof is 10 m long. Find its volume.

Find the volume of a cylinder with a radius of 2 cm and a height of 10 cm.

A pipe is 10 cm long, with outer radius 3 cm and inner radius 2 cm. Find the volume of material.

Two 1 cm cubes are glued face to face. What is the surface area of the new object?

An L-shaped prism has a front face with an area of 50 cm² and is 8 cm long. Find its volume.

More practice: warm-up

Find the volume of a rectangular prism 5 cm × 4 cm × 3 cm.

Find the area of a circle with a radius of 5 cm.

Find the volume of a cylinder with a radius of 1 cm and a height of 10 cm.

What kind of units measure volume?

A triangular prism has a triangle face with an area of 12 cm² and a length of 5 cm. Find its volume.

More practice: on your own

A shed is a box 3 m wide, 2 m deep and 2 m tall, with a triangular prism roof. The roof's triangle has a base of 2 m and a height of 1 m, and the roof runs the 3 m width. Find the shed's volume.

A grain bin is a cylinder with a radius of 4 m and a height of 6 m. Find its volume.

A 10 cm wooden cube has a round hole with a radius of 2 cm drilled straight through it. What volume of wood is left?

A 4 cm cube is glued to the middle of the top of a 10 cm cube. Find the surface area.

A tent is a triangular prism 2 m long. Its triangle has a base of 1.6 m and a height of 0.6 m. Find the slant side, then the area of fabric for the whole tent, including the floor.

Challenge

A water trough is half of a cylinder lying on its side, with a radius of 30 cm and a length of 150 cm. How many litres can it hold?

A round concrete planter is a hollow cylinder 40 cm tall. Its outside diameter is 60 cm and its inside diameter is 50 cm (the bottom is open). How much concrete is in it?

Which holds more: a 10 cm cube, or a cylinder with a 10 cm diameter and a 12 cm height? Explain.

Check your understanding

Quiz

To find the volume of a composite object made of two parts, you:

What is the volume of a cylinder with a radius of 3 cm and a height of 10 cm?

Two 1 cm cubes are glued face to face. What is the surface area?

A triangular prism has a triangle with base 4 cm and height 3 cm, and it is 10 cm long. What is its volume?

A pipe is 10 cm long with outer radius 2 cm and inner radius 1 cm. What volume of material is it made of?

When you find the surface area of a composite object, what do you do with faces where the parts are joined?

For parents and teachers

Big idea: The Ontario curriculum (E2.3) asks Grade 8 students to solve problems involving the volume and surface area of composite three-dimensional objects, using appropriate formulas. Your child already knows the formulas for prisms and cylinders from Grade 7; the new skill is combining them and spotting hidden faces.

Common difficulties:

Including hidden faces in the surface area. Ask, "Could you paint this face if the object were glued together?"
Forgetting to subtract a hole.
Using the diameter instead of the radius in circle formulas.
Unit conversions: 1 m³ = 1 000 000 cm³, not 100 cm³.

How to help: Build the composite object from boxes and cans together. Real projects work well too: "How much paint for the shed?", "How much soil for this planter?" This lesson also builds A1 skills: planning a multi-step solution and organizing work clearly.