Mathematics Magazine for Grades 1-12
Regular (Platonic) Solids
Plato felt that
everything in this world was a mere shadow of perfection. The tree you are
looking at is not perfect, a broken branch here, peeling bark there, but it is a
reflection of a perfect tree, a platonic tree, in the spiritual world. We
still retain the adjective platonic, though it is generally restricted to
If the chosen n-gon
is a hexagon or higher, three interior angles sum to 360° or more, hence they
can't fit together to make a corner. Platonic solids are based on the
triangle, square, or pentagon.
Let's start with
the triangle. If three triangles meet at a corner then a fourth triangle
completes the shape. This is called a tetrahedron, 4 faces, 6 edges, 4
Let four triangles
meet at each corner. This is called an octahedron, 8 faces, 12 edges, 6
Let five triangles
meet at each corner. This is called an icosahedron, 20 faces, 30 edges, 12
If 6 or more
triangles meet at a point, the angles sum to 360° or more, so lets move on to
Let three squares
meet at each corner. This is called a cube, 6 faces, 12 edges, 8 vertices.
Let three pentagons
meet at each corner. This is called a dodecahedron, 12 faces, 30 edges, 20
Solutions from the Previous Issue:
ABCDE is regular pentagon and DEFG is a square. Find the
measures of angle EFA and angle DAF.
Let n the number of the sides of a regular polygon. The angles of a regular polygon having n sides are:
and “<A” means the angle A
<DEF = 90o
<AED = 108o
The triangle AEF is isoscele.
<FEA = 360 o –
(<DEF +<DEA)= 360 o – (90 o +108o) =162o
The triangle AEF is isosceles and <EFA = <EAF = [180 o –(<FEA)]/2=
(180 o - 162o )/2= 9o
<DAF = <DAE + <EAF (1)
The triangle AED is an isosceles triangle. That means that < EAD = <EDA = (180 o -108 o)/2 = 36 o
From (1) we have <DAF =36
o + 9o = 45 o
2. For each of the numbers: 41, 83, 32, the first digit is greater in value than the second digit. How many 2-digit numbers have this property?
Solution: If we begin to list the numbers in groups:
we can see that the total number of 2-digit numbers, for which the first digit is greater than the second digit, will be 1 + 2 + ... + 9 = 45.
How many 3-digit numbers exist
for which the first digit is greater in value than both the second digit and the
3. A cone is designed so that it fits perfectly into a cylindrical container.
Given the volume of the cone is 100 cm3 and the curved surface area of the cylinder is 150 cm2, what is the height of the container?
Volume of cone = 1/3 πr2h = 100 πr2h = 300.
Curved surface area of cylinder = 2πrh = 150.
=>r = 4 cm