Rectangular Solids and Prisms Revision Notes for Leaving Cert Mathematics

Rectangular Solids and Prisms

What is a rectangular solid?

A rectangular solid (also called a cuboid) is a three-dimensional shape where all faces are rectangles. Think of a shoebox or a brick - these are perfect examples of rectangular solids.

Key formulas for rectangular solids

For any rectangular solid with length $l$ , breadth $b$ , and height $h$ :

Volume formula: $\text{Volume} = l \times b \times h$

Surface area formula: $\text{Surface area} = 2lb + 2lh + 2bh$

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The surface area formula works because a rectangular solid has six faces that come in three pairs of identical rectangles. Each pair contributes twice its area to the total surface area.

What are prisms?

A prism is a three-dimensional shape that has the same cross-sectional area throughout its length. When you imagine slicing through a prism at any point along its length, you always get the same shape.

Types of prisms

Prisms can have different cross-sectional shapes:

Rectangular prisms (like rectangular solids)
Triangular prisms (with triangular cross-sections)
Cylindrical prisms (with circular cross-sections)
Irregular prisms (with any other uniform cross-section)

Volume formula for prisms

The volume formula for any prism is:

$\text{Volume of prism} = \text{Area of cross-section} \times \text{Length}$

Or more simply: $\text{Volume} = A \times l$

Where $A$ is the area of the cross-section and $l$ is the length of the prism.

Working with composite cross-sections

Sometimes a prism has a cross-section made up of multiple shapes combined together. To find the total cross-sectional area, you calculate the area of each individual shape and add them together.

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Worked Example: Composite Prism

A prism is 30 cm long. Its cross-section consists of a rectangle (40 cm × 10 cm) with a triangle (base 40 cm, height 10 cm) on top.

Step 1: Calculate the rectangular area Area of rectangle = 40 × 10 = 400 cm²

Step 2: Calculate the triangular area Area of triangle = ½ × 40 × 10 = 200 cm²

Step 3: Find total cross-sectional area Total area = 400 + 200 = 600 cm²

Step 4: Calculate volume Volume = 600 × 30 = 18,000 cm³

Nets of 3D shapes

A net is a two-dimensional shape that can be folded to create a three-dimensional shape. When you unfold a cardboard box completely flat, you're looking at its net.

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Key Facts About Nets:

Every 3D shape has at least one possible net
Some shapes (like cubes) have multiple different net configurations
A cube has 11 different possible nets
To find the surface area of a shape, you can calculate the total area of its net

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Worked Example: Volume from a Net

From the net shown, we can identify this will fold into a rectangular box with dimensions 5 cm × 3 cm × 2 cm.

Volume calculation: Volume = length × breadth × height Volume = 5 × 3 × 2 = 30 cm³

Additional Worked Examples

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Worked Example: Triangular Prism

Calculate the volume of a triangular prism where the triangular base is equilateral with sides of 3 cm each, and the prism length is 6 cm.

Step 1: Find the area of the equilateral triangle For an equilateral triangle with side length 3 cm: Height = $\frac{\sqrt{3}}{2} \times 3 = \frac{3\sqrt{3}}{2}$ cm Area = $\frac{1}{2} \times 3 \times \frac{3\sqrt{3}}{2} = \frac{9\sqrt{3}}{4}$ cm²

Step 2: Calculate volume Volume = Area of cross-section × length Volume = $\frac{9\sqrt{3}}{4} \times 6 = \frac{27\sqrt{3}}{2}$ cm³ ≈ 23.4 cm³

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Worked Example: L-shaped Prism

Calculate the volume of an L-shaped prism with the dimensions shown.

Step 1: Break down the L-shape into two rectangles

Rectangle 1: 7 cm × 3 cm = 21 cm²
Rectangle 2: 5 cm × 2 cm = 10 cm²

Step 2: Find total cross-sectional area Total area = 21 + 10 = 31 cm²

Step 3: Calculate volume (assuming length is 4 cm from the diagram) Volume = 31 × 4 = 124 cm³

Exam Tips

When tackling problems involving rectangular solids and prisms, remember these essential strategies:

Always identify what type of shape you're dealing with first
For composite shapes, break them down into simpler parts
Double-check your units - volume is always in cubic units (cm³, m³, etc.)
When working with nets, visualise how they fold to understand the 3D shape

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Common Mistakes to Avoid:

Forgetting to add areas when dealing with composite cross-sections
Mixing up length, breadth, and height in volume calculations
Using incorrect units (area units for volume, or vice versa)
Not showing all working steps clearly

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Key Points to Remember:

Volume of rectangular solid: $V = l \times b \times h$
Volume of any prism: $V = \text{cross-sectional area} \times \text{length}$
A net is a 2D shape that folds into a 3D shape
For composite cross-sections, add the areas of individual shapes
Always check your units and show clear working steps

Rectangular Solids and Prisms (Leaving Cert Mathematics): Revision Notes