Topic 11: Nets, Areas and Volumes of Solids

By the end of this topic, you will be able to:

(a) form nets of common solids.

(b) identify common solids and their properties including faces, edges and vertices.

(c) state units of measure.

(d) convert units from one form to another.

(e) calculate the surface areas of three-dimensional figures.

(f) calculate the volume of cubes and cuboids.

Keywords

• cone
• cube
• cuboid
• cylinder
• prism
• pyramid
• areas

Introduction

Some of you have metallic boxes for keeping your items in the dormitories or even at home. All of you are familiar with card boxes that are used for packing items like soap, biscuits etc. These are examples of solids. In this topic, you will learn Nets, Areas and Volumes of Solids, develop the skills of drawing and making two- and three-dimensional shapes and explore their properties. Knowledge of these properties will enable you to solve problems involving solids.

11.1 Forming nets of common solids 1.

Activity 11.1 Forming nets of common solids

1. On a cardboard or manila paper, accurately draw the diagram shown below.

2. Cut out the drawn figure and fold it clearly to form a solid. You may apply glue or cello tape to join the edges.

(a) What shape do you obtain?

(b) Form nets of other different common shapes you know on a manila sheet of paper and display them in class.

11.2 Identifying common solids and their properties

Activity 11.2 Identifying properties of common solids

Carefully study the shape you formed in

Activity 11.2. Use it to answer the following questions.

1. How many faces, edges and vertices does the shape have?

2. Share your findings with the rest of the class.

11.3 Stating Units of Measures

In Topic 10, you calculated the area of shapes like; triangles, rectangles and squares. How did you obtain the units of the area? Let us look at the following examples to discover how to obtain units of area.

Examples

1. A rectangular plot of land measures 10 metres by 16 metres. What is the area of the land?

Solution

The plot of land is rectangular:

So, area of rectangle = length x width Thus,

i.e. multiply figures together and units together = 160 m2

2. The teacher’s table top measures 120 cm by 55 cm. Find the area of the table top.

Solution

Area = Length x width

= 120 cm x 55 cm

= 6600 cm2

Learning point

The units of the result are a product of the units for the quantities that are being multiplied.

Exercise 11.3

1. Peter and Charity were asked to determine the area of the chalkboard in their classroom. Peter measured the length and recorded 3 m while Charity measured the height of the chalkboard and recorded 150 cm. Calculate the area of the chalkboard.

2. The figure below shows an isosceles triangle whose total area is 0.0135 m2.

11.4 Converting units from one form to another

In Topic 10, you measured the length of objects in various units, including centimetres (cm) and metres (m). Also in Senior One Physics, you carried out conversions of units of length from one form to another, for example, from cm to m and vice versa. In this section, you will deal with the conversion of units of area from one form to another.

Activity 11.4 Converting from m2 to cm2 and vice versa

1. Measure and record the dimensions of a science laboratory table top in:
   • (a) metres to one decimal place.
   • (b) centimetres to one decimal place.
2. Calculate the area of the table top in each case.
3. Are the two values of the area related in any way?
4. Share your work with the rest of the class.

Learning point

Converting the units of derived quantities like area from say m2 to cm2 is best done by first converting the units of length to the required units, then calculating the area.

Examples

1. A book cover measures 25 cm by 20 cm. Find the area of the book cover in: (a) cm2. (b) m2. 2.
2. Convert the following as instructed.
   • (a) 100 cm2 to m2
   • (b) 0.55 m2 to cm2

Solutions

1. (a) Area = Length x Width

(b) Area = 25 cm x 20 cm

= 500 cm2 = Length x Width

= 0.25 cm x 0.20 cm

= 0.05 m2 100

2. (a) 100 cm2 = m2 10000

= 0.01m2

(b) 0.55 m2 = 0.55 x 10000 cm2

= 5500 cm2

Exercise 11.4

1. In an activity to measure the dimensions of a chalkboard, Jane and James recorded the following results

(a) Who among the two learners who recorded the correct values?

(b) Calculate the area of the chalkboard according to

(i) Jane.

(ii) James.

(c) Do you think the two values obtained in (i) and (b)(ii) are related in any way?

2. Peter would like to lay floor tiles in his bedroom which measures 4.5 m by 3.6 m. He goes to the hardware shop to place an order for tiles and finds tiles measuring 12 cm by 24 cm. How many tiles will he need to buy so they are enough for his bedroom? She intends to distribute it equally to her three children.

3.Teacher Beatrice owns consolidated land measuring 2.4 km much does each get in m2.

4. A sunflower plant is 1.8 m tall. Over the next month, it grows a futher 34 cm. How tall is the sunflower plant at the end of the month?

5. Peter is a long-distance runner. In a training session he runs around the 400 m tracks 23 times. He wanted to run a distance of 10 km. How many more times does he need to run around the track to achieve this?

6. A water cooler comes with water containers that hold 12 I of water. The cups provided for use each hold 150 ml. A company estimates that each of its 20 employees will drink 2 cups of water a day. How many 12 I bottles will be needed for each working week (Monday to Friday)?

11.5 Calculating surface areas of three-dimensional figures

Activity 11.5 Calculating surface areas of three-dimensional figures

1. You are provided with an empty box in the shape of a cuboid.
   • (a) Measure and record the dimensions of the box.
   • (b) Suggest how the surface area of the box can be obtained.
   • Use the suggested method in (b) to calculate the total surface area of the box.
2. Present your work to the class.

Exercise 11.5

1. Tom has a bedroom in the form of a cuboid with a flat ceiling and a square floor. The inside of the room measures 5.5 m by 5.5 m by 4.5 m and has a window and a door measuring 1 m by 1.5 m and 1 m by 2.5 m respectively. Tom decides to cover the walls and the ceiling with sheets of newspapers. How many full sheets of paper measuring 25 cm by 30 cm will he need?

2. Ms. Edvine keeps her household items in a metallic case which measures 100 cm by 56 cm by 40 cm. She decides to change the colour of the metallic case by painting. At the hardware shop, the shopkeeper tells her every small tin of Weather Guard paint costs UGX 15,000 and can fully paint an area of 5,920 cm2. Establish how much money Edvine will need to purchase enough tins to paint the case fully.

3. Giving your answers correct to 3 significant figures, calculate the total surface area of each of the following cylinders.

11.6 Calculating the volume of cubes and cuboids 1.

Activity 11.6 Calculating the volume of cuboids

1. Study the figures below and answer the questions that follows.

1. (a) How many cubic units are in one layer?
2. (b) How many layers filled the figure?
3. (c) What is the total volume of the whole figure?
4. Present your findings to the rest of the class.

Activity 11.7 Finding more about the volume of cuboids

1. Carefully study the diagram below and use it to answer the following questions. Smallest unit

(a) What shape is the figure? Give reason(s) for your answer.

(b) What shape is the smallest unit of the figure?

(c) How much space does the smallest unit of the figure occupy?

(d) Establish how many smallest units make up the entire figure, hence state the volume of the figure.

2. Present your ideas to the class.

Activity 11.8 Calculating volume of cubes

1. Carefully study the figure below and answer the questions that follow.

(a) What shape is the figure?

(b) What shape is the smallest unit?

(c) How many smallest units make up the whole figure? Present your ideas to the class.

Exercise 11.6

1. A fish pond in the form of a cuboid is 210 cm deep. The pond is 12 m long and 6.6 m wide. How much water will be required to fill the pond?

2. Ms. Dian is constructing a water tank for harvesting rainwater from her house at home. The base of the tank is rectangular and measures 2.5 m by 3.5 m. How tall should the tank be constructed if she needs a tank that can hold a maximum of 52500000 cm3 of water?

3. Calculate the volume of

(a) a cuboid of dimensions 3 cm by 5 cm by 7 cm.

(b) a cube of side 1.2 m.

4. Find the amount of water that can fill the containers shown in the following figures.