1. 4cm
5cm
9cm²
5cm
4.5cm
2.5cm

2. What’s the same and what’s different??
Discuss…
What’s the same and what’s different??

3. The volume of a solid is the amount of space occupied by the solid.
The greater the volume of a solid the more space it takes up.
The volume of a solid is the number of unit cubes that the solid can be divided into.
2 units
1 unit
1 unit
1 unit
4 units
24 unit cubes altogether
3 units

4. This cuboid is made up of centimetre cubes (cm³). What is its volume?

5. What is the volume of this cuboid?
5 cm
2 cm
10 cm
100 cm³

6. What is the volume of this cuboid?

7. Answers
5 x 2 x 7 = 70 cm3
3 x 2 x 8 = 48 cm3
5 x 6 x 2 = 60 cm3
2 x 2 x 6 = 24 cm3
5 x 2 x 3 = 30 cm3
2 x 7 x 4 = 56 cm3
7 x 7 x 2 = 98 cm3
2 x 4 x 5 = 40 cm3
2 x 9 x 2 = 36 cm3
Extension question:
54 ÷ (9 x 2)
= 54 ÷ 18
= 3cm

8. Josh is having a birthday party
Josh is having a birthday party! He wants to make boxes of sweets to give to his guests.
He starts with a square piece of paper that measures 20cm x 20cm. How big should the squares be that he cuts out of the corners to maximise the volume of the box?

9. Record your results in the table provided.
Don’t forget your units of measurements!

10. Extension questions
What if the square you cut out doesn’t have to use whole number measurements (e.g. 2.4cm)? Would this change your answer?
What if you started from a 10cm x 20cm rectangle instead? What would the biggest volume be?

11. Surface areas of cubes and cuboids
What is surface area?
width
Think about finding the area of a square or rectangle…
height
Area = height x width
What about if we phrase surface area differently…
The area of the surface

12. Surface areas of cubes and cuboids
The area of the surface
How could we find the surface area of a cuboid using the height, width and length? W L
How many faces does a cuboid have? H 6
So we could add together the areas of all 6 faces!

13. Surface area = Front Back Left Right Top Bottom (Length x Height) +W
Back
Top
Left
Surface area =
Front
Right
Front
Back
Left
Right
Top
Bottom
(Length x Height) +
(Height x Width) +
(Length x Width) +
(Length x Width)
Bottom
By adding the area of all of the faces, we can find the surface area of the whole cuboid.

14. Answers
38cm² 168cm²
62cm² 68cm²

15. A cube is cut out of a larger cube and stuck into the corner, as shown
A cube is cut out of a larger cube and stuck into the corner, as shown. What is the surface area of the resulting shape?
All lengths are in centimetres

16. HINT: There are 15 faces!

17. Total Surface Area
= 115cm²

18. Calculate the areas of the following shapes
Starter
Calculate the areas of the following shapes
3cm
4cm
5cm
4cm
3cm
5cm
6cm

19. Calculate the areas of the following shapes
Starter
Calculate the areas of the following shapes
3cm
15cm²
4cm
16cm²
5cm
4cm
3cm
5cm
15cm²
28.26cm²
6cm

20. Cross Section – The shape of the slice
Prism – A shape that has the same cross section all the way through

21. A prism is a 3-D shape which has the same cross-section throughout its height.
Triangular prism
Pentagonal prism
Cuboid

22. Identify the prisms

23. = area of cross-section x vertical height
Volume of a prism
= area of cross-section x vertical height
Cross-section
Vertical height

24. = area of cross-section x vertical height
Volume of a prism
= area of cross-section x vertical height
Example: Find the volume of this prism
25 x 7
= 175
cm³
25 cm2
7 cm

25. = area of cross-section x vertical height
Volume of a prism
= area of cross-section x vertical height
Example: Find the volume of this prism
Area of cross-section = ½ x 6 x 3
= 9
cm²
3 cm
10 cm
Volume = 9 x 10
= 90
cm³
6 cm

26. Checkpoint
100mm³
72cm³ 2
90cm³
120m³
5cm 5m 6m
250cm³

27. Volume of Prisms
Thoughts and crosses
Calculate the volumes of 4 of the prisms, either vertically, horizontally or diagonally

28. The area of the surface
How could we find the surface area of a triangular prism using the height, width, depth and slant height? S H D
How many faces does a triangular prism have? W 5
So we could add together the areas of all 5 faces!

29. S Surface area = H Front Back Left Right Bottom (Width x Height ÷ 2) +
(Height x Depth) +
(Slant x Depth) +
(Width x Depth) D W
Bottom
By adding the area of all of the faces, we can find the surface area of the whole triangular prism.

30. Answers
36cm² 240cm²
352cm² 372cm²

31. Calculate the areas and circumferences of these circles to 1 d.p.
Starter
Calculate the areas and circumferences of these circles to 1 d.p. Q1 Q2 Q3
2.5 cm
7 cm
8 cm Q4 Q5 Q6
3 cm
4.5 cm
2 cm

32. Answers Q1 A = 38.5 cm², C = 22.0 cm Q2 A = 19.6 cm², C = 15.7 cm

33. Volume of a prism = area of cross section x length
Calculate the volume of this cylinder. Give your answer to 1 d.p.
Area of cross-section =
π x 12²
= … cm²
Volume =
… x 20
= cm³
Calculate the volume of this cylinder. Give your answer to 3 s.f.
Area of cross-section =
π x 5²
= … cm²
Volume =
78.539… x 12
= 943 cm³

34. Answers
150.80cm³
18.85cm³
192.42cm³
cm³
150.80cm³
192.42cm³
cm³
18.85cm³
2.27cm
2.90cm
5.27cm
5.57cm

35. πd h Circumference = πd Height (h)
Surface area of curved part of cylinder = πdh

36. Surface area of cylinder = 2πr² + πdh
Area of top circle = πr²
Surface area of curved part of cylinder = πdh
Area of bottom circle = πr²
Surface area of cylinder = 2πr² + πdh

37. Calculate the total surface area of the cylinder, giving your answer to 1 d.p.:
Top =
π x 2²
= … cm²
4cm
Curved =
π x 4 x 6
= … cm²
Bottom =
π x 2²
= … cm²
Total =
12.566… … …
6cm
= cm²

38. 192 cm²
cm²

39. Answers
175.93cm²
186.92cm²
43.98cm²
633.03cm²
175.93cm²
186.92cm²
43.98cm²
633.03cm²
19.06cm
28.92cm
111.75cm
63.90cm

40. Starter
Calculate the volume of the cuboid and the area of the circle. Give your answers to 3 s.f. where appropriate.
94 cm
9 m
89 cm
140 cm
cm³
or 1.17 m³
63.6 m²

41. Volume of Pyramids
Volume of pyramid = x base area x vertical height
Calculate the volume of the pyramids below:
1 3 x 20 x 13 x 12
= 1040 cm³
1 3 x 7 x 4 x 6
= 56 cm³

42. Volume of Pyramids
Volume of pyramid = x base area x vertical height
A cone is a circular-based pyramid! Calculate the volume and give your answers to 1 d.p.
6 mm
8 cm
11 mm
5 cm
1 3 x π x 5.5² x 6
= mm³
1 3 x π x 5² x 8
= cm³

43. Answers
70 cm3
cm3
12 cm3
m = 12.5 cm
n = 1.31 cm

44. Answers
70 cm3
cm3
12 cm3
m = 12.5 cm
n = 1.31 cm

45. ? cm
3 cm
4 cm 6 5 7 4

46. ? cm
3 cm
4 cm 6 5 7 4

47. 9 cm
6 cm
? cm
6.7 12
10.8
13.6

48. 9 cm
6 cm
? cm
6.7 12
10.8
13.6

49. 8 cm
? cm
3 cm
7.4
8.5
9.3
6.6

50. 8 cm
? cm
3 cm
7.4
8.5
9.3
6.6

51. Surface Area of Cones Surface area of cone = πr² + πrl l h r
Curved surface area (you get given this bit in exams!)
Area of base r
Calculate the surface area of this cone. Give your answer to 1 d.p.
(π x 4²) + (π x 4 x 9)
= cm²
9 cm
4 cm

52. Surface Area of Cones Surface area of cone = πr² + πrl l h r
Curved surface area (you get given this bit in exams!)
Area of base r
Sometimes, we need to calculate l first.
l =
= 7.211… cm
6 cm
(π x 4²) + (π x 4 x 7.211…)
= cm²
4 cm

54. Volume of a sphere = 4/3 x π x r³
Surface area of a sphere = 4 x π x r²
Calculate the volume and surface area of the sphere below:
Volume = 4/3 x π x 8³
= cm³
8cm
Surface area = 4 x π x 8²
= cm² 54

55. Answers 1) a) V = 113.1cm³, SA = 113.1cm²
b) V = 904.8cm³, SA = 452.4cm²
c) V = cm³, SA = cm² d) V = cm³, SA = 615.8cm²
e) V = 179.6cm³, SA = 153.9cm² f) V = 47.7cm³, SA = 63.6cm²
2) a) cm² b) cm³
3) V = cm³, height = 125cm