C hristmas Problem Solving
GCSE Shapes and Space
Question 1
HINT
Ahmed has a roll of wrapping paper 2m long and 75cm wide. He has to wrap
the following presents, does he have enough paper? Show your workings.
Maze-in-a-ball
The maze-in-a-ball
can be wrapped with
a rectangular piece of
paper
Speakers X2
12.5cm
10cm
SPHERE
Radius – 10cm
CUBES
Side length - 12.5cm each
Pyramid of chocolate
Bubble bath gift set
22.5cm
30cm
23cm
27cm
SQUARE BASED RIGHT PYRAMID
Square base side length – 27cm
Slant height – 22.5cm
TRIANGULAR PRISM
Triangle base (equilateral) side length – 23cm
Prism height – 30cm
Extension
Does Ahmed have enough paper to wrap each present
in a single sheet (of any shape)?
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Question 2
Julie is putting 3 presents – a
jumper, a pair of socks and
a teddy bear – into 3 gift
boxes. The boxes are cubes
of different sizes.
Box 1
Box 2
Box 3
x cm
y cm
z cm
Use the information in the table to answer the questions:
Volume of object
Clues
Jumper – 11,000 cm3
Box 1 can hold the socks but nothing else.
Teddy Bear – 50,000 cm3
Box 2. There are three ways to select two presents.
Two of these combinations fit into Box 2.
Socks – 750 cm3
Box 3 can hold all three items without being full but could not hold an
extra jumper in addition to the first three items.
2b) Use your answer to part 1 to write down an inequality
2a) For each box write down an
describing the range of possible values of:
inequality to describe the range of
values the volume of the box could
i) x
ii) y
iii) z
take.
Question 3
120 cm
Megan and her Dad have bought a Christmas
tree and want to take it home in the boot of
their car. The tree is 170cm tall.
First they assume that the car boot is a cuboid.
73 cm
106 cm
Will the tree fit? Show your working.
Extension
The wheel arches of the car mean that the boot is actually a trapezoidal prism. As the wheel arches
take up 5cm each side, will the Christmas tree still fit?
View from above
120 cm
120 cm
73 cm
5 cm
106 cm
5 cm
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5 cm
Question 4
When Megan and her dad get the Christmas tree home, they need to rearrange their
furniture to fit it in the living room. Use the clues to place all the furniture into the room.
Clues:
Sofas
• The rug has 3 whole objects on
top of it, including the dog
x2
• Other than what is on the rug, no
objects touch each other
Coffee
tables
Television
Lamp
• Objects can be against the wall,
but must not touch the window or
the radiator
Foot
stool
Tr
ee
• There must be enough room to
open and close the door
4a) When all the furniture is in the
room estimate how much floor
has been covered. Give your
answer in units2.
Rug
Dog
Window
1 unit2
Radiator
Do
o
rw
ay
Extension
Exactly how much of the floor has
been covered?
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Solutions
Question 1
Yes – all the objects can be wrapped with a single roll of paper (200cm x 75cm).
Present
Shape of paper
Surface area of wrapping paper (Rounded up to 1 decimal place)
Maze-in-a-ball
Rectangle
31.41…cm × 62.83…cm = 1974cm²
Speakers
Cube net X2
6 × 12.5cm² = 937.5cm² EACH (or 1562.5cm² if wrapped together)
Pyramid of chocolate
Pyramid net
27² + 4 × (½ × 27 × 22.5) = 1944cm²
Bubble bath gift set
Triangular prism net
2 × (½ × 23 × 396.75) + 3 × (23 × 30) = 2528.2cm²
Total = 8321.2cm² (or 8008.7cm²)
The sheets of paper to cover each present can be cut from the roll of paper as below*
Maze-in-a-ball
Pyramid of
chocolate
Bubble bath gift set
Speakers
* Not to scale
Question 2
2a 750 cm3 ≤ volume of box 1 < 11000 cm3
50 750 cm3 ≤ volume of box 2 < 61 000 cm3
61 750 cm3 < volume of box 3 < 72 750 cm3
2b9.09 ≤ x < 22.24
37.02 ≤ y < 39.36
39.53 < z < 41.75
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Question 3
Extension
1202 + 1062 = 14400 + 11236 = 25636
1152 + 1062 = 13225 + 11236 = 24461
( 25636 = 160.11246 = 160 3sf)
( 24461 = 156.40012 = 156 3sf)
732 + ( 25636)2 = 5329 + 25636 = 30965
732 + ( 24461)2 = 5329 + 24461 = 29790
30965 = 175.96874 = 175cm
(rounded down to nearest cm)
29790 = 172.597798 = 172cm
(rounded down to nearest cm)
Yes, the tree will fit
Yes, the tree will still fit.
Question 4
There are multiple solutions.
One solution is shown here:
In this solution, the exact amount of floor covered is 130 + 5π
Lamp
≅ 145.71 units2
Tr
ee
Sofa
Coffee
table
Rug
Sofa
Window
Coffee
table
Sofa
Radiator
or
wa
y
Dog
Do
Television
Foot stool
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