Mathematics Magazine for Grades 112 



TheoryRegular (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
relationships. If the chosen ngon
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
vertices. Let four triangles
meet at each corner. This is called an octahedron, 8 faces, 12 edges, 6
vertices. Let five triangles
meet at each corner. This is called an icosahedron, 20 faces, 30 edges, 12
vertices. If 6 or more
triangles meet at a point, the angles sum to 360° or more, so lets move on to
squares. 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
vertices. Solutions from the Previous Issue:
1.
ABCDE is regular pentagon and DEFG is a square. Find the
measures of angle EFA and angle DAF.
Solution:
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 = 90^{o } <AED = 108^{o } The triangle AEF is isoscele. <FEA = 360^{ o} – (<DEF +<DEA)= 360^{ o} – (90^{ o} +108^{o}) =162^{o } The triangle AEF is isosceles and <EFA = <EAF = [180^{ o} –(<FEA)]/2= (180^{ o}  162^{o })/2= 9^{o} <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 }+ 9^{o }= 45^{ o
} 2. For each of the numbers: 41, 83, 32, the first digit is greater in value than the second digit. How many 2digit numbers have this property? Solution: If we begin to list the numbers in groups: 10; 20,21; 30,31,32; 40,41,42,43; ... ; 90,91,92,93,94,95,96,97,98 ; we can see that the total number of 2digit numbers, for which the first digit is greater than the second digit, will be 1 + 2 + ... + 9 = 45. How many 3digit numbers exist
for which the first digit is greater in value than both the second digit and the
third digit? 3. A cone is designed so that it fits perfectly into a cylindrical container.
Given the volume of the cone is 100 cm^{3} and the curved surface area of the cylinder is 150 cm^{2}, what is the height of the container? Solution: Volume of cone = ^{1}/_{3} πr^{2}h = 100 πr^{2}h = 300. Curved surface area of cylinder = 2πrh = 150. =>r = 4 cm Proposed Problems 1 2 3 4 


