Perimeter, Area, and Volume Revision Notes for HSC SSCE Mathematics Standard

Surface Area and Volume of Composite Solids

What is a composite solid?

A composite solid is a three-dimensional shape formed by joining together two or more basic shapes. These basic shapes might include cubes, cylinders, cones, spheres, prisms, or pyramids. Understanding how to work with composite solids is essential because many real-world objects are combinations of simpler shapes rather than single, basic forms.

When working with composite solids, you'll need to calculate either their surface area (the total area of all outer surfaces) or their volume (the amount of space they occupy). The methods for these calculations build on your knowledge of formulas for basic shapes, but require careful visualisation and systematic working.

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Composite solids appear frequently in real-world applications, from architectural designs to everyday objects. Mastering these calculations helps you solve practical problems involving complex three-dimensional shapes.

Surface area of composite solids

Finding the surface area of a composite solid follows the same fundamental approach used for right prisms and pyramids. The key is to break down the problem into manageable parts by identifying each individual surface, calculating its area, and then combining these areas appropriately.

Method for calculating surface area

Follow these steps systematically when finding the surface area of a composite solid:

Step 1: Visualise the surfaces Picture all the faces that make up the outer surface of the solid. It can help to sketch the solid from different angles or mentally rotate it to see all sides.

Step 2: Calculate each face area Write down the appropriate formula for each surface and calculate its area. You may need formulas for squares, rectangles, circles, triangles, or other shapes depending on the solid.

Step 3: Add the areas Sum up the areas of all individual surfaces to find the total surface area.

Step 4: Check your work Verify that you've counted all surfaces and haven't missed any faces or counted any twice.

Step 5: Write your answer Express your final answer with the correct units (square metres, square centimetres, etc.) and to the specified level of accuracy.

chatImportant

Common mistake: The most frequent error when calculating surface area is forgetting to count all surfaces or counting the same surface twice. Always double-check your work by visualising the solid from all angles.

Worked example: Surface area with a cylindrical hole

Let's examine a real-world example involving a composite solid where material has been removed.

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Worked Example: Finding the Surface Area of a Composite Solid

Problem: A red cube sculpture features a cube with a cylindrical hole removed through its centre. The cube has a side length of $8.53$ m and the cylindrical hole has a radius of $1.42$ m.

a) Find the surface area of one face with the cylindrical hole (correct to two decimal places).

b) Calculate the total surface area of the red cube sculpture, including only the outer red surfaces (correct to two decimal places).

Solution:

Part a) - Area of face with hole

The face we need consists of a square with a circular hole removed. This means we calculate the area of the square, then subtract the area of the circle.

For a square: $A = s^2$

For a circle: $A = \pi r^2$

Face area = Square area - Circle area

$A = s^2 - \pi r^2$

Substituting the values:

A = 8.53^2 - \pi \times 1.42^2

A = 72.7609 - \pi \times 2.0164

A = 72.7609 - 6.3348...

A = 66.4261... \approx 66.43 \text{ m}^2

The area of one face with the hole is 66.43 m².

Part b) - Total surface area

Now we need to identify all the surfaces of this composite solid:

Two faces with cylindrical holes (the top and bottom faces we calculated in part a)
Four faces that are complete squares (the four vertical sides of the cube)

The cylindrical hole creates an internal curved surface, but the question asks only for the outer red surfaces, so we don't include the inside of the cylindrical hole.

Total surface area:

SA = (2 \times 66.43) + (4 \times 8.53^2)

SA = 132.86 + (4 \times 72.7609)

SA = 132.86 + 291.0436

SA = 423.9036... \approx 423.90 \text{ m}^2

The surface area of the red cube sculpture is 423.90 m².

infoNote

Exam tip: When a solid has holes or cut-outs, remember to subtract the area of the removed shape from the relevant face. Also consider whether any internal surfaces (like the inside of a hole) should be included in your calculation - read the question carefully!

Volume of composite solids

To find the volume of most common solids, we use standard formulas. A composite solid combines two or more of these basic shapes, so calculating its volume requires identifying the components, finding each individual volume, and then combining these volumes by addition or subtraction.

Method for calculating volume

Use this systematic approach when finding the volume of a composite solid:

Step 1: Divide into common solids Break down the composite solid into two or more basic shapes (such as cylinders, cones, cubes, rectangular prisms, or spheres) that you know how to work with.

Step 2: Calculate each volume Use the appropriate formula to find the volume of each basic component. Common formulas include:

Cylinder: $V = \pi r^2 h$
Cone: $V = \frac{1}{3}\pi r^2 h$
Rectangular prism: $V = l \times w \times h$
Sphere: $V = \frac{4}{3}\pi r^3$

Step 3: Combine the volumes Add the volumes together if the shapes are joined. Subtract volumes if one shape has been removed from another (like a hollow section).

Step 4: Write your answer State your final answer with the correct units (cubic metres, cubic centimetres, etc.) and to the required level of accuracy.

chatImportant

Key concept: When shapes are joined together, you add their volumes. When one shape is removed from another, you subtract the volumes. Always identify which operation is needed before you begin calculating.

Worked example: Volume of a grain silo

Let's work through a practical example involving a composite solid made by joining two shapes together.

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Worked Example: Finding the Volume of a Composite Solid

Problem: A grain silo consists of a cylinder with a cone positioned on top. The cylinder has a diameter of $16$ m and a height of $20$ m. The cone has the same diameter as the cylinder (so its base sits perfectly on top of the cylinder) and a height of $2$ m. Calculate the volume of the silo correct to one decimal place.

Solution:

This composite solid combines two basic shapes: a cone and a cylinder. We need to calculate each volume separately, then add them together.

Cone volume:

The formula for the volume of a cone is:

$V = \frac{1}{3}\pi r^2 h$

First, find the radius. Since diameter $= 16$ m, the radius is:

$r = \frac{16}{2} = 8 \text{ m}$

The cone's height is $h = 2$ m.

Substituting into the formula:

V_{\text{cone}} = \frac{1}{3}\pi r^2 h

V_{\text{cone}} = \frac{1}{3} \times \pi \times 8^2 \times 2

V_{\text{cone}} = \frac{1}{3} \times \pi \times 64 \times 2

V_{\text{cone}} = \frac{128\pi}{3}

V_{\text{cone}} = 134.0412866... \text{ m}^3

Cylinder volume:

The formula for the volume of a cylinder is:

$V = \pi r^2 h$

Using the same radius ( $r = 8$ m) and the cylinder's height ( $h = 20$ m):

V_{\text{cylinder}} = \pi r^2 h

V_{\text{cylinder}} = \pi \times 8^2 \times 20

V_{\text{cylinder}} = \pi \times 64 \times 20

V_{\text{cylinder}} = 1280\pi

V_{\text{cylinder}} = 4021.238597... \text{ m}^3

Total volume:

Add the volume of the cone to the volume of the cylinder:

V_{\text{total}} = V_{\text{cone}} + V_{\text{cylinder}}

V_{\text{total}} = 134.0412... + 4021.238...

V_{\text{total}} = 4155.2798... \approx 4155.3 \text{ m}^3

The volume of the grain silo is 4155.3 m³.

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Exam tip: When working with composite solids, always draw a sketch and label all given measurements. Identify clearly which measurements belong to which component shape. Keep your calculator in the correct mode (radians or degrees as needed) and don't round intermediate values - only round your final answer.

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Key Points to Remember:

A composite solid is formed by combining two or more basic shapes (such as cylinders, cones, cubes, or prisms).
For surface area, visualise all the outer surfaces, calculate each face's area using appropriate formulas, then add them together. Don't forget to check you've counted every surface!
For volume, break the composite solid into basic components, calculate each volume separately, then add or subtract these volumes as needed.
When a shape has been removed (like a hole or cut-out), subtract that area or volume from the main shape.
Always include proper units in your final answer (m² for area, m³ for volume) and round to the specified level of accuracy only at the end of your calculation.

Perimeter, Area, and Volume (HSC SSCE Mathematics Standard): Revision Notes

Surface Area and Volume of Composite Solids

What is a composite solid?

Surface area of composite solids

Method for calculating surface area

Worked example: Surface area with a cylindrical hole

Volume of composite solids

Method for calculating volume

Worked example: Volume of a grain silo

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