Problem 1
1 yard = 91.44 cm (  1 m)
1 yard = 3 feet  1 feet = ………… cm
1 feet = 12 inches  1 inch = ………… cm
1 mile = 1760 yards  1 mile = ………… m = ………… km
“More joy of mathematics”, by Theoni Pappas
Problem 2
Can you prove it too ?
“The joy of mathematics”, by Theoni Pappas
Problem(s) 3 : Squaring the circle and irrational numbers
1. A circle has a radius of 1 m.
We want to construct a square that has the same area as this circle : what should be the side of
this square ?
Same area
2. Calculate the diagonal of a square of side length 1 m.
Problem 4
Let ABC be an isosceles triangle, such as : 𝐴𝐵 = 10 cm, ABC = 36°.
1. Calculate the two other angles of this triangle.
2. Let 𝐻 be the midpoint of 𝐵𝐶.
a. Find the length 𝐵𝐻, and infer the length 𝐵𝐶. Give the results to 2 decimal places.
b. Work out the height 𝐴𝐻.
c. Calculate the area of the triangle 𝐴𝐵𝐶.
Problem 5 : What’s the best way to cut a cake ?
1. How would you cut a circular cake into five equal pieces ? eight equal pieces ?
2. Let’s suppose now that the cake is square, and you need to cut it into a number of identical
pieces for your guests.
Rob Eastaway and Jeremy Wyndham, in their book “Why do buses come in threes”, give the
following solution :
“There is a slightly eccentric method that works
regardless of how many people you are cutting for. All
you need is to [divide] the perimeter into equal lengths.
If your number of guests is seven, for example, mark out
the perimeter of the square into seven equal lengths.
Now locate the centre of the cake and make cuts from the
perimeters marks to the middle as shown. The seven
pieces are of equal volume and have the same amount of
icing […].”
Here’s the beginning of their proof :
1. What is the base of
each of the five triangles
on the right ?
2. What is their height ?
3. Do these triangles
really have the same
area ? What is this
common area ?
1 yard = 91.44 cm (  1 m)
1 yard = 3 feet  1 feet = ………… cm
1 feet = 12 inches  1 inch = ………… cm
1 mile = 1760 yards  1 mile = ………… m = ………… km
Problem 6 : 3D
H
G
F
E
D
C
A
I
𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻 is a cuboid, such as 𝐴𝐵 = 20, 𝐴𝐷 = 𝐴𝐸 = 7 , and 𝐼 is the midpoint of 𝐴𝐵.
1. Work out the lengths 𝐷𝐼 and 𝐻𝐼.
2. Are the straight lines 𝐼𝐻 and 𝐼𝐺 perpendicular ?
B