2.2 Space and shape  (Page 3/3)

Polygons are closed figures with three or more straight sides. If all the sides are the same length, and all the internal angles are equal, we call them regul a r polygons. Triangles are three-sided polygons, and an equil a ter a l triangle is a regul a r three-sided polygon. A square is a regular four-sided polygon. Pent agons have five sides, hexagons have six sides and heptagons have seven. Make a list of as many of these special names as you can find.

Here are several closed plane figures. Decorate them and write the name of each polygon on the shape.

B. Investig a tion

Choose four polygons from the group above, all regular, but with four different numbers of sides. Now measure the sizes of the internal angles of each. Try to find out whether it is possible to make a formula to tell you how large the angles are, and what they add up to.

The following table will be helpful. As you can see, there are infinitely many polygons.

• The characteristics in the table above are needed when you have to decide how to tile a floor (say) with regular polygons so that they don’t overlap and don’t leave gaps. Some of these polygons will work alone, and some can or must be combined.
• Design and draw a repeating tiling pattern of your own, using only regular polygons and colour it so that the pattern shows clearly.

C. Three-dimension a l closed figures

• If these shapes have sides made up of polygons, then we call them polyhedr a. A regul a r polyhedron has faces that are congruent regular polygons, with internal angles the same shape and size.
• In contrast to the polygons, there are only five regular polyhedra. They have been known since the time of Plato and the Greek mathematicians; this is why they are known as the five Platonic Solids.

D. Project

Research the five Platonic Solids, finding their names and properties, and other interesting deductions and facts about them. Make an attractive poster or models of these solids showing the facts associated with each. Below are pictures of the five solids.

Assessment

Memorandum

Discussion

• This guide includes two pages of figures for constructing simple right prisms. Photocopy enough for the learners to make at least two of the figures. It would be best if the copies could be made on very light card (or heavy paper). If they are asked to colour some of the parts (e.g. the base and top) it might make it easier to explain some of the more difficult formulae.

• The two formulae for right prisms are, in general:
• Total Surface Area = double the base area + height of prism × perimeter of base
• Volume = base area × height of prism
• Ensure that learners are clear on the units (squared or cubed) appropriate to each formula.
• Another difficulty that learners might encounter is that the word height is used in calculating the area of triangles as well as being one of the dimensions of right prisms. A useful trick is to use h for the triangle case and H for the prism case.
• Breaking down the steps required for the calculations is a useful method for learners who get confused by the components in the formula. Of course, very competent learners will substitute values straight into the formula. This is an effective system, and should be encouraged where appropriate.

Solutions – exercise:

Rectangular prism: TBO = 412 cm ² Vol = 480 cm ³

Triangular prism: TBO = 307,71 cm ² Vol = 360 cm ³

Cylinder: TBO = 402,12 cm ² Vol = 603,19 cm ³

Granny’s Jam Pot: Vol = 8 595,40 cm ²

11 Square–based jars: Vol = 8 096 cm ²

11 Rectangle–based jars: Vol =8 633,63 cm ²

So, granny must use the rectangular–based jars if she wants to fit all the jam in!

3 = triangle; 4 = tetragon; 5 = pentagon; 6 = hexagon; 7 = heptagon; 8 = octagon; * = not polygon

TEST 1

1. Explain how you would recognise a right prism.

2. Explain how you could find the base of a right prism.

3. Calculate the total surface area and the volume of each of the following three prisms. Give your answers accurate to two decimal places.

TEST 1 – Memorandum

1. Essential points in the explanation: three-dimensional; top and base congruent plane shapes; side(s) at right angles to base.

2. Any reasonable explanation, e.g. if the chosen side is placed at the bottom, the description of a right prism fits what you see.

3. Rectangular right prism: TBO = 1 939,68 cm ² Volume = 5 769,72 cm ³

Triangular right prism: TBO = 1 507,74 mm ² Volume = 2 312 mm ³

Cylinder: TBO = 8 022,37 m ² Volume = 41 593,67 m ³