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Mathematics Magazine for Grades 1-12  

 

Grade 9

     
Theory

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 relationships. 

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 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 = 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:

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 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 third digit?
Can you generalize for n-digit numbers?
What about 3-digit numbers for which the first digit is greater than the sum of the second and third digits?

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?

Solution:      

Volume of cone = 1/3 πr2h = 100                  πr2h = 300.

Curved surface area of cylinder = 2πrh = 150.

=>r = 4 cm

Proposed Problems

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