Maths Investigation: Drainage Channels.

/2A = 573cm²

So the area of the semicircle is 573cm²

Triangles

Example 1

30cm 30cm

As there are two sides to a triangle the 60cm length must be split in half, so each side is now 30cm long. But there can also be a variety of angles, ranging from 0º to 90º

Here is the area for 30º

Sin15º = x/30

X = 0.26 x 30

X = 7.76

Cos15º = h/30

H = 28.89

Area = 7.76 x 28.89 = 224.88cm²

Example 2

Sin10º x X/30

X = 0.174 x 30

X = 5.21

Cos10º x h/30

H = 0.98 x 30

H = 29.54

Area = 5.21 x 29.54 = 154.0597cm²

As you can see this is a long drawn out process, so if we were to create a table it would save having to draw all the triangles out. This is on the next page

Rectangles

X X

60 - 2X

Example 1

5cm 5cm

50cm

Area = 50 x 5

Area = 250cm²

Graph of results on next page.

Trapeziums.

20cm 20cm 20cm 20cm

20cm

Opp xcm

Adj Hyp

H 50º 20cm

Sin50º = x/20

X = 20sin50

Cos50º = h/20

H = 20cos50º

Instead of doing this drawn out process I found a formula that worked, it is

Area = (a+b/2) x c

B

C

When Changing the

A angle change these

numbers.

Area = 20 +(20 + 20sin50 + 20sin50)/2 = Ans

Area = Ans x 20sos50

So to find a formula where you don't have to draw out every shape again and again we use algebra.

X 20 X

S Z H H Z S

B X

H Z S

SinZ = x/s

X = s x sinzº

Cosz = h/s

H = S x cosz

Area = b + (b + 2s sinzº)/2 = Ans

Area = Ans x 20coszº

There are graphs on the next few pages to represent these results.

(To enter these results into a spreadsheet I needed to know about radians which is how a spreadsheet works.)

2 x 3.142 = 360º

º = 3.142/180º

N Sides

180/n

60/n

36º

Side = 60/n

Angle = 180/n

180/n

90/n

A H

O 30/n

TOA CAH SOH

(Tan 90/n = 30/n) /h

h = (30/n) / tan 90/n

Area = 30/n x (30/n) /tan 90/n

= 30/n x 30/n tan 90/n

Formula for Area is = 900/ (n x tan 90/n)