# Directed Investigation Essay

1410 words - 6 pages

INTRODUCTION
In the present day world, many schools and educational institutes burden students with the memorisation of multiple surface area formulas for a particular prism. It is vital to have the understanding of how various surface area formulas make geometry appear a hard stream of mathematics. The aim of this directed investigation is to discuss the topic question “Is it possible to develop a general formula for the surface area of any prism” and furthermore to develop a formula that can be used as a shortcut for the surface area of any prism calculations. This investigation will look into finding the relationship between the surface area of different prisms by using a whole range of ...view middle of the document...

The 4 prominent figures were-

Equilateral triangle – based prism
NET
SKETCH

Square prism
NET SKETCH

Regular Hexagonal based prism
NET SKETCH

Regular octagonal based prism
NET SKETCH

From figure 1.1 it was indicated that the middle section of each net was a rectangle. Every prism has flat sides and the same cross section all along its length. A cross section is the shape that is produced after cutting straight across through a prism. The laws of mathematics explain that if a prism does not have a rectangular middle section. Therefore the shape of the middle section of each net was a rectangle.

The shape of each middle section was considered as a rectangle, therefore, the area of the middle section of each net can be calculated using the formula a = l*w. Through the formula the area of each middle section can be calculated to -
AREA OF THE MIDDLE SECTION OF EACH NET
PRISM AREA ( A = L*b)
720 cm2
960 cm2
1440 cm2
1920 cm2

Workout
a. Prism type –
Area = area of first rectangle * the no of sides

b. Prism type –
Area = area of the first rectangle * the no of sides

c. Prism type –
Area = area of first rectangle * the no of sides

d. Prism type –
Area = area of the first rectangle * the no of sides

The above formula worked efficiently to determine the area of the middle sections, however, this formula led to the development of an alternative formula that could be replaced by the area formula to find the area of the middle section. The new formula developed was –

Area of the middle section = length/height of the prism * the perimeter of the regular polygon base
OR
A = H*P

The development of the new formula proved to be easier to use, because it involved less steps and working out. It can be said that this was the first hint to the new formula that was to be discovered.

PART 2
In order to continue progressing from part 1, the next step taken into considerations was to draw the net of a right angled triangle prism of height 20 cm with the side lengths of 6, 8 and 10 cm and to draw the net of a rectangular prism of height 20 cm with the dimensions of 11 cm * 15 cm.

Rectangular Prism
NET SKETCH

Triangular Prism
NET SKETCH

There were a range of patterns observed between the nets of prisms sketched in part 1 and part 2. It was observed that the prisms in part 1 had regular polygon bases causing the middle section dimensions to be same, whereas the prisms in part 2 had irregular polygon bases causing the rectangular sides in the middle section to be different. The only component similar about the prisms in two parts was the dimensions of the two bases.

To find the area of the middle section of the prims drawn above. The formula used was

AREA OF THE MIDDLE SECTION OF THE NEW NETS (FIGURE 1.3)
PRISM AREA (A = H*P)
Triangular prism 480 cm2
Rectangular prism 1040 cm2

Workout

a. Prism type – Triangular prism
Area = Height * the perimeter of the base

b. Prism type –...

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