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- END UNIT 1 ASSIGNMENTOpened: Saturday, 27 May 2023, 12:00 PMDue: Monday, 29 May 2023, 12:00 AM
UNIT 13:Calculating the circumference of a circle and the volume of cuboids and cubes
13.1 The circumference of a circle
Activity 13.1
Study the diagrams below.
(i) Trace the circular path on each of them.
(ii) Which objects with circular paths have you brought?
(iii) Wrap a string around the circular path of the object. Untie the
string and measure its length using a ruler.
(iv) Discuss your findings.
Tip:
The distance around a circular object is called the circumference.
What are the circumferences of the objects you measured in Activity 13.1?
Practice Activity 13.1
1. Identify circular objects in the school compound.
2. Measure their circumferences.
3. Why do you think these objects are circular? Discuss.
13.2 Finding pi (
)
Look at the circle below.
The distance AB is diameter of the circle.
The diameter of a circle is a straight line which passes through its centre.
Activity 13.2
Have the following materials:
Rulers, tape measure, string, circular objects, manila papers
Steps
(i) Measure the diameter of the circular objects you have, as shown
below. The distance must pass through the centre of the circle.
Note: Half of the diameter is called the radius.
(ii) Prepare a chart on a manila paper as shown in the table below.
(iii) Now measure the circumference of the objects you have. Record the
circumference and diameter of each object on a chart.
Tip: The result of: circumference ÷ diameter is called pi. The symbol for
pi is
13.3 Calculating the circumference of a circle
Let us remember the parts of a circle as shown below. Discuss and name
the parts labelled a – d.
Practice Activity 13.2
C. Discuss and find the answers to the following questions. Present your
findings.
1. The diameter of a circular ring is 21 cm. What is its circumference?
2. A bicycle wheel has a diameter of 98 cm. What is the circumference
of the wheel?
3. The diameter of the circle at the centre of a football field is 9.8 m.
(a) What is the circumference of the circle?
(b) A player ran round the circle three times. What distance did
he cover?
Practice Activity 13.3
2. The radius of a circle is 15 cm. What is its circumference?
3. The radius of the lid of a bucket is 45 cm. Calculate the circumference
of the lid. Discuss your steps to answer.
4. The radius of a circular water tank is 3 1/2
m. A rope is tied around the
tank. What is the length of the rope? Explain how you arrived at your
answer.
Practice Activity 13.4
13.4 Cubes and cuboids
Activity 13.5
• Discuss and identify the length, width and height of different boxes.
• Prepare a chart like one shown below.
• Measure the length, width and height of different boxes.
• Record your results in your chart.
• In which boxes are
(a) the length, width and height equal? Explain.
(b) the length, width and height different? Justify your results.
Tip: The boxes whose three sides are equal are called cubes.
The boxes whose three sides are different are called cuboids.
13.5 Properties of cubes and cuboids
Activity 13.6
Study the cuboid below
Make a chart like the one below on manila paper
Take the cubes and the cuboids in turns. Count their vertices, faces and
edges. Record them in your chart. Discuss your results.
Activity 13.7
Make a chart as shown below.
Study cubes and cuboids then fill in your chart appropriately.
(a) Which of the boxes are cubes? Justify your answers.
(b) Which of the boxes are cuboids? Justify your answers.
(c) Discuss the similarities between cubes and cuboids.
(d) List the difference between cubes and cuboids
Practice Activity 13.5
1. How many vertices does a cube have?
2. How many edges are there in a cuboid?
3. How many faces are there in a cube?
4. How many faces are there in cuboid?
5. What is the product of edges and vertices in a cube?
6. Observe the shapes of objects in the classroom.
(a) Which ones are cubes? Why are they cubes?
(b) Which ones are cuboids? Explain why they are cuboids.
13.6 Nets of cubes and cuboids
Activity 13.8
• Take a box. Open it as shown below.
• The completely opened shape of the box is called a net. To make a
cube or a cuboid, we must first prepare a ‘net’.
• Open other cubes and cuboids to form different nets.
• How many faces are in each net?
• Fold a net to form a cube or cuboid.
• Fold nets to form a cube or cuboid. Present your model.
Making nets
Activity 13.9
Have the following materials: manila paper, pair of scissors, ruler, glue.
On manila paper, draw the net of a cuboid. Its measurements should be
length 10 cm, width is 8 cm and height is 6 cm.
Cut out the net from the manila paper. Make sure the net has flaps
which are about 1 cm wide.
Fold the net to make a cuboid. Make sure the edges are neatly folded
along the lines.
Practice Activity 13.6
1. Make the following cubes and cuboids.
(a) Length 10 cm, width 10 cm and height 10 cm.
(b) Length 20 cm, width 15 cm and height 10 cm.
You need a pencil, a ruler, manila papers, a pair of scissors and glue.
2. Draw the following on manila paper. Cut the nets out. Fold them along
the dotted lines.
(a) Which of the nets will make a cube?
(b) If the net does not make a cube, explain reasons why you think
it does not.
3. Draw the shapes below using the given measurements.
Cut the shapes out. Fold them along the dotted lines.
(a) Which of the nets make cuboids?
(b) Why do the other nets NOT make cuboids? Explain your
answers.
(c) How would you re-arrange some of the nets to make cuboids?
13.7 Calculating the volume of cubes and cuboids
Volume of cubes
Activity 13.10
Cut out square pieces of manila paper with sides of 1 cm each.
• Observe the space occupied by one square card. The area of the square
card is 1 cm 2.• Stack the square card up to a height of 1 cm.
• Take the cubes and cuboids that you made from the previous activities.
Which ones occupy bigger space? Which ones occupy less space?
• Compare the space occupied by your exercise book to that occupied
by the text book.
• Discuss the space occupied by various objects in the classroom.
Activity 13.11
• Make several cubes like one shown below. This is a unit cube.
• Make a layer like the one below using unit cubes.
(i) How many cubes are there along the length?
(ii) How many cubes are there along the width?
(iii) How many cubes are there along the height?
(iv) Count the number of cubes in the layer.
(v) Calculate the number of cubes in the layer.
• By adding similar layers on top, make the following:
(i) How many cubes are along the length?
(ii) How many cubes are along the width?
(iii) How many layers are in the stack? Explain.
(iv) How many cubes form each stack? This is the volume of
the stack. Discuss your results.
(v) Now let us calculate the volume as follows:
Volume = cubes along length × cubes along width × cubes
along height.
Tip:
From Activity 13.11, above, in stack (a);
Its length = 4 cm
Its width = 4 cm
Its height = 4 cm
Its volume = length × width × height
= 4 cm × 4 cm × 4 cm = 64 cm 3
Now, calculate the volume of cuboid (b) using the formula.
Volume = length × width × height
Example 13.4
Find the volume of the diagrams below.
(c) A rectangular box is 65 cm long, 40 cm wide and 28 cm high. Calculate
the volume of the box.
Solution
(a) Volume = length × width × height
= 6 cm × 6 cm × 6 cm = 216 cm
(b) Volume = length × width × height
= 38 cm × 21 cm × 15 cm = 1 1970 cm 3
(c) Volume = length × width × height
= 65 cm × 40 cm × 20 cm = 72 800 cm 3
Practice Activity 13.7
Calculate the volume of each of the following.
5. A rectangular tank measures 4.3 m long, 2.4 m wide and 1.5 m high.
Calculate the volume of the tank. Justify your answer.
6. A large carton measures 64 cm long, 32 cm wide and 30 cm high. What
is the volume of the carton? Where do we use a carton? Discuss.
7. The figure below represents a water tank. If it is filled with water,
what is the volume of the water in it in cubic meters? Explain how you
arrived at your answer.
8. A cube has 18 cm sides. What is its volume when it is a quarter full?
9. The figure below represents a water tank. It was half filled with water.
How much water was in it? Discuss your steps.
10. The figure below represents a swimming pool. How much water in m 3
are in it when it is half full? Explain how you found your answer. Tell
importance of a swimming pool.
Example 13.5
A box is 30 cm long, 20 cm wide and 15 cm high. What is the volume of
the box?
Volume = length × width × height = l × w × h
= (30 × 20 × 15) cm 3
= 9 000 cm 3
Practice Activity 13.8
1. What is the volume of a cube whose sides are 20 cm. Explain how you
arrive at the answer.
2. A rectangular water tank (cuboid) measures 4 m long, 3 m wide and
2 m high. What is its volume?
3. A building brick is 20 cm long, 15 cm wide and 8 cm high. What is its
volume? Discuss your steps.
4. A box is 35 cm long, 22 cm wide and 18 cm high. Calculate 23 of its
volume. Present your answer.
5. An underground tank is 8 m long, 6 m wide and 10 m high. How much
water in m 3 is required to fill it? Explain how you arrive at your answer.
13.8 Finding one dimension of a cuboid
Activity 13.12
Playing a game – The missing dimension
Required items: – 48 cubes each with 5 cm sides.
– Chart like the one below.
Play this game.
Arrange unit cubes along the given sides.
Example: From (a) arrange as below.
Arrange such layers to have 36 unit cubes. How many layers are there?
That is the number of cubes along the height.
Repeat similar steps for (b) to (d). Fill in the missing blanks. Play in turns.
Find out the shortest method to get the missing dimension. Present yourmethod to the class.
Example 13.6
(a) The volume of a cuboid is 420 cm 3. It has a length of 10 cm and width
7 cm. What is its height?
Solution
Volume = l × w × h
420 cm 3= 10 cm × 7 cm × h
420 cm 3 = 70 cm 2 × h
420 cm 3 ÷ 70 cm 2= h
The length is 6 cm.
Note: height = volumelength × width
(b) Study the cuboid below. Its volume is 4 536 cm 2.
Find its width.
Solution
V = length × width × height
4 536 cm 3 = 21 cm × w × 12 cm
4 536 cm 3 = 21 cm × 12 cm × w
4 536 cm 3= 252 cm 2 × w
4 536 cm 3 ÷ 252 cm 2 = w
4536
252 cm = w
18 cm = w
width = 18 cm
Note: width = volumelength × height
(c) A rectangular tank has a volume of 7 m3. It is 2 m wide and 1.4 mhigh. Find the length of the tank.
Practice Activity 13.9
Copy and complete the table below.
5. A carton has a volume of 142 560 cm 3. Its width is 36 cm and its height
is 72 cm. Form an equation and calculate its length.
6. The volume of a rectangular water tank is 414 720 cm 3. Its length is
144 cm while its width is 36 cm. Calculate its height. Explain how you
arrive at your answer.
7. A log of wood is in the shape of a cuboid. It is 35 cm long and 30 cm
high. Its volume is 25 200 cm 3. How wide is the log?
8. A container is 40 cm wide and 18 cm high. Its volume is 21 600 cm 3.
What is its length?
9. A box has a volume of 160 m 3. It is 8 m long and 5 m wide. What is its
height?
10. A water tank has a length of 3.8 m, 2.5 m wide. Its volume is 38 m 3.
What is its height?
13.9 Find the height of a cuboid given its volume andbase area
Activity 13.13
Collect several unit cubes. Use them to make different cuboids. In turns,
player 1 makes a cuboid and states how many cubes are in it. Player 2
states how many unit cubes are in one layer. Player 3 states how many
layers make the height.
Each correct answer given is awarded 3 marks. A wrong answer is not
awarded any mark.
The cuboid is dismantled. Player 2 makes his or her own cuboid.
Player 3 gets the cubes in one layer. Player 1 gets the layers along theheight.
Compare: Volume ÷ base and height.
What do you notice? Discuss your findings.
Discuss and match the following.
Justify your answer.
Example 13.7
The volume of the cuboid is 72 000 cm 3. Its base area is 2 400 cm 2. Whatis its height?
Practice Activity 13.10
Copy and complete the table below.
4. The volume of a cuboid is 18 000 cm 3. Its base is a square of side 30 cm.
What is
(a) its base area? Explain how you got answer.
(b) its height?
5. The base of a tank is a rectangle whose length is 90 cm and width
50 cm. Its volume is 10 000 cm 3. Calculate its height.
6. The volume of a rectangular water tank is 8 000 cm 3. It has a base
area of 160 cm 2. What is its height? Explain your answer.
7. The floor of a classroom measures 8 m long and 7 m wide. The volume of
the classroom is 168 m 3. What is the height of the classroom. Compare
this with the height of your classroom. Discuss your results.
8. To make a brick, a mason used 11 220 cm 3 of mortar. He made a brick
whose length was 34 cm and width 22 cm. Calculate the height of the
brick.
9. The volume of an underground tank is 84 m 3. Its base area is 28 m 2.How deep is the tank?
13.10 Finding the area of a face of a cuboid
Activity 13.14
Study the following cuboids. Discuss and find the area of their shaded
faces. The volume of each cuboid is 1 728 cm 3. The shaded face becomes
the base and the given length is the height.Remember base area × height = volume.
Present your findings and how you got your answer to the class.
Example 13.8
Calculate the area of the shaded face of the cuboid below. Its volume is2 618 cm 3
Solution
Volume of the cuboid = 2 618 cm 3, length = 17 cm.
Area of shaded face = volume
length
= 2 618 cm 3
17 cm= 154 cm 2
Practice Activity 13.11
1. Find the area of the shaded faces of the given cuboids. The volume (V)
and one dimension have been given in each case.(a) V = 3 120 cm 3 (b) V = 1 428 cm 3
2. Study the figures below:
(i) Calculate the area of the shaded part.
(ii) Discuss and find the missing dimension of each figure.
(iii) Explain how you calculate the missing dimension.
3. A carpenter made a rectangular wooden box with a volume of 1.44 m 3.
Its length was 1.5 m and width 0.8 m.
(a) What is its area? Discuss your steps.
(b) Find its height.(c) How did you get its height? Explain.
Revision Activity 131. Collect a flexible stick that is 154 cm long.
Fold it into a circle to form a wheel. Use tape to attach the ends.
(a) What is the circumference of the wheel made from the flexible
stick?
(b) What is the diameter of the wheel. (Use a ruler to measure)
2. What is the circumference of a wheel whose diameter is 28 cm?
Take π = 22
7.
3. What is the circumference of the circle below. Take π = 227 .
4. How many faces are there in an open cube?
5. What is the product of the edges and the vertices of a cuboid?
6. The following nets are folded to form cubes or cuboids. Indicate the
ones that will form:(i) cubes (ii) cuboids (iii) neither cuboids nor cubes
7. To make a roundabout, the contractor needed a circumference of
88 m. What diameter does he use to get the circumference? Explain
your answer.
8. Small bottles packed in small packets measuring 10 cm long, 10 cm
wide and 20 cm high were packed in a carton measuring 80 cm long,
60 cm wide and 40 cm high. How many bottles fit in the box? Explain
your answer.9. What is the volume of the cuboid below.
10. What is the volume of a cube whose sides are 18 cm? Explain how
you arrive at your answer.
11. Study the container for the truck below. It is for carrying loads likecartons of books.
The container measures 7 m long, 3 m wide and 4.2 m high. Find the
volume of the container. Justify your answer.
12. The volume of a cube is 64 cm 3. What is the measurement of one of
its sides? Present the process of arriving at the answer.
13. The volume of a box is 405 m 3. It is 15 m long and 9 m wide. What
is its height? Discuss your steps.14. The diagram below represents a box made up of cardboard.
Its volume is 4 530 cm 3. What is the area of its top? Explain how you
get the answer.
15. Find the length of a cuboid whose volume is 87 360 cm 3. It is 48 cm
wide and 35 cm high.
16. The diagram below represents a water tank. Its volume is 32 m 3.
What is its depth?
Present the process of getting the answer.
17. A rectangular container had two faces made from metal sheets.
The shaded part represents the metal sheets. Its volume is
19 285 cm 3. Calculate the area of the metal sheet used to make the
shaded parts.Explain your answer.
Word list
Diameter Radius Circumference
Cube Cuboid Net
Circle Base area Height
Width Length
Task
Do the following.
(i) Read each word aloud to your friend.
(ii) Write the meaning of each of the words above. Discuss with your friend.(iii) Write sentences using each of the words above. Read with your friend.