Properties and Volumes of Three-Dimensional Figures for JSS 1

Three dimensional Figures

Performance Objectives

By the end of the lesson, students should be able to:

1. identify the properties of cubes and cuboids;
2. identify the properties of pyramids and cones;
3. identify the properties of cylinders and spheres;
4. find the volumes of cubes and cuboids.

Meaning of Three-Dimensional Figures

A three-dimensional (3-D) figure is a solid shape that has length, width and height.

Unlike two-dimensional shapes such as squares and rectangles which are flat, three-dimensional figures occupy physical space. Examples of three-dimensional figures found in homes and schools include boxes, tins of milk, footballs and conical party hats.

To understand the properties of these figures, three key terms must be defined: faces, edges and vertices.

A face is a flat or curved surface of a solid shape.

An edge is a line segment where two faces of a solid shape meet.

A vertex is a point where three or more edges of a solid shape meet.

The plural form of vertex is vertices.

Properties of Cubes and Cuboids

Cubes and cuboids are common six-sided solid shapes. While they share the same number of faces, edges and vertices, their faces have different shapes.

The Cube

A cube is a three-dimensional solid shape bounded by six equal square faces.

The properties of a cube include:

1. It has 6 flat faces, and all the faces are equal squares.
2. It has 12 straight edges, and all the edges are equal in length.
3. It has 8 vertices.

The Cuboid

A cuboid is a three-dimensional solid shape bounded by six rectangular faces.

The properties of a cuboid include:

1. It has 6 flat faces, which are rectangular in shape.
2. Opposite faces of a cuboid are equal in size and shape.
3. It has 12 straight edges, which are not all equal in length.
4. It has 8 vertices.

Properties of Pyramids and Cones

Pyramids and cones are solid shapes that taper to a single point at the top. This top point is called the apex.

The Pyramid

A pyramid is a solid shape with a polygon base and triangular faces that meet at a common point at the top.

Pyramids are named after the shape of their bases. A square-based pyramid is the most common type studied at this level.

The properties of a square-based pyramid include:

1. It has 5 flat faces (1 square base and 4 triangular side faces).
2. It has 8 straight edges.
3. It has 5 vertices (4 vertices at the base and 1 apex at the top).

The Cone

A cone is a solid shape with a circular base and a curved surface that tapers to a point at the top.

The properties of a cone include:

1. It has 2 faces (1 flat circular face which is the base, and 1 curved surface).
2. It has 1 curved edge around the circular base.
3. It has 1 vertex at the top, which is the apex.

Properties of Cylinders and Spheres

Cylinders and spheres are curved solid shapes that do not have straight edges or sharp corners.

The Cylinder

A cylinder is a solid shape with two parallel circular bases of equal size and one curved surface connecting them.

The properties of a cylinder include:

1. It has 3 faces (2 flat circular faces at the ends, and 1 curved surface).
2. It has 2 curved edges.
3. It has 0 vertices.

The Sphere

A sphere is a perfectly round three-dimensional solid shape where every point on the surface is equal in distance from the centre.

The properties of a sphere include:

1. It has 1 curved surface.
2. It has 0 flat faces.
3. It has 0 edges.
4. It has 0 vertices.

Summary of Three-Dimensional Figure Properties

The table below provides a quick reference for the number of faces, edges and vertices of the six three-dimensional figures discussed.

Volumes of Cubes and Cuboids

Volume is the amount of three-dimensional space occupied by a solid object.

Volume is measured in cubic units, such as cubic centimetres ($\text{cm}^3$) or cubic metres ($\text{m}^3$).

Volume of a Cube

Since all the edges of a cube are equal in length, the volume is found by multiplying the length of the side by itself three times.

The formula for the volume of a cube is:

$$V = l \times l \times l = l^3$$

Where:

1. $V$ is the volume of the cube.
2. $l$ is the length of one side of the cube.

Worked Example 1 (Cube)

Calculate the volume of a cube with a side length of $5\text{ cm}$.

Solution:

State the formula:

$$V = l^3$$

Substitute the given value into the formula:

$$V = 5\text{ cm} \times 5\text{ cm} \times 5\text{ cm}$$
$$V = 125\text{ cm}^3$$

The volume of the cube is $125\text{ cm}^3$.

Worked Example 2 (Cube)

A cubic water tank has a side length of $2\text{ m}$. Find the volume of the tank.

Solution:

State the formula:

$$V = l^3$$

Substitute the given value into the formula:

$$V = 2\text{ m} \times 2\text{ m} \times 2\text{ m}$$
$$V = 8\text{ m}^3$$

The volume of the water tank is $8\text{ m}^3$.

Worked Example 3 (Cube)

Find the volume of a small wooden cube whose edge measures $1.5\text{ cm}$.

Solution:

State the formula:

$$V = l^3$$

Substitute the given value into the formula:

$$V = 1.5\text{ cm} \times 1.5\text{ cm} \times 1.5\text{ cm}$$
$$V = 2.25 \times 1.5\text{ cm}^3$$
$$V = 3.375\text{ cm}^3$$

The volume of the wooden cube is $3.375\text{ cm}^3$.

Volume of a Cuboid

The volume of a cuboid is calculated by multiplying its length, width and height together.

The formula for the volume of a cuboid is:

$$V = l \times w \times h$$

Where:

1. $V$ is the volume of the cuboid.
2. $l$ is the length of the cuboid.
3. $w$ is the width of the cuboid.
4. $h$ is the height of the cuboid.

Worked Example 1 (Cuboid)

Find the volume of a cuboid with a length of $10\text{ cm}$, a width of $6\text{ cm}$ and a height of $4\text{ cm}$.

Solution:

State the formula:

$$V = l \times w \times h$$

Substitute the given values into the formula:

$$V = 10\text{ cm} \times 6\text{ cm} \times 4\text{ cm}$$
$$V = 60 \times 4\text{ cm}^3$$
$$V = 240\text{ cm}^3$$

The volume of the cuboid is $240\text{ cm}^3$.

Worked Example 2 (Cuboid)

A rectangular classroom measures $8\text{ m}$ in length, $6\text{ m}$ in width and $3.5\text{ m}$ in height. Calculate the volume of air in the classroom.

Solution:

State the formula:

$$V = l \times w \times h$$

Substitute the given values into the formula:

$$V = 8\text{ m} \times 6\text{ m} \times 3.5\text{ m}$$
$$V = 48 \times 3.5\text{ m}^3$$
$$V = 168\text{ m}^3$$

The volume of air in the classroom is $168\text{ m}^3$.

Worked Example 3 (Cuboid)

A chalk box has a length of $12\text{ cm}$, a width of $8\text{ cm}$ and a height of $10\text{ cm}$. What is the volume of the chalk box?

Solution:

State the formula:

$$V = l \times w \times h$$

Substitute the given values into the formula:

$$V = 12\text{ cm} \times 8\text{ cm} \times 10\text{ cm}$$
$$V = 96 \times 10\text{ cm}^3$$
$$V = 960\text{ cm}^3$$

The volume of the chalk box is $960\text{ cm}^3$.

How to Measure and Calculate the Volume of a Cuboid Box

This practical activity describes how to measure a physical cuboid object and calculate its volume using a standard measuring instrument.

The required materials for this activity are:

1. A rectangular box (such as a shoe box, matchbox or small carton).
2. A standard metric ruler (30 cm) or a measuring tape.
3. A pen and a sheet of paper for recording measurements.

The measurement and calculation procedure consists of the following steps:

1. Place the rectangular box on a flat, stable surface.
2. Identify the three dimensions of the box: length (the longest horizontal side), width (the shorter horizontal side) and height (the vertical side from the table to the top of the box).
3. Align the zero mark of the ruler with one corner of the box along its length. Measure the length to the nearest millimetre and record the value in centimetres.
4. Align the zero mark of the ruler along the width of the box. Measure and record the value in centimetres.
5. Align the zero mark of the ruler vertically along the height of the box. Measure and record the value in centimetres.
6. Multiply the three recorded values together using the formula $V = l \times w \times h$.
7. Write down the final calculated volume, ensuring the unit is written as cubic centimetres ($\text{cm}^3$).

Evaluation Exercises

Section A: Properties of 3-D Figures

Solve the following problems to test understanding:

1. State the number of faces, edges and vertices of a cube.
2. How does a cuboid differ from a cube in terms of its faces?
3. Identify the solid shape that has 1 flat circular face, 1 curved surface, 1 curved edge and 1 vertex.
4. State the number of vertices in a cylinder and a sphere.
5. A square-based pyramid has how many triangular faces?

Section B: Volume Calculations

Solve the following problems to test understanding:

1. Calculate the volume of a cube with a side length of $7\text{ cm}$.
2. A cubic box has a side length of $12\text{ cm}$. Find its volume.
3. Find the volume of a cuboid with a length of $15\text{ cm}$, a width of $8\text{ cm}$ and a height of $5\text{ cm}$.
4. A rectangular container is $10\text{ m}$ long, $4\text{ m}$ wide and $3\text{ m}$ deep. What is the volume of the container?
5. A block of wood is in the shape of a cuboid. It has a length of $20\text{ cm}$, a width of $10\text{ cm}$ and a height of $6\text{ cm}$. Calculate its volume.