Question

A long time ago Dulani found an island shaped like a triangle with three straight shores of length 3km, 4km and 5km. He said nobody could come within 1km of his shore. What was the area of his exclusion zone?

Answer 1

You basically have this triangle (which is the island itself)

Answer 2

then you add on this buffer zone of 1 km around the entire island

Answer 3

so you need to find the area of this red zone

you can break it up into 3 rectangular pieces (shown in blue in this new attachment) and 3 circular sectors (shown in green)

Answer 4

do you see how to get the answer?

Answer 5

So like for the area of one of the rectangular pieces, do we just add 1km to each sides?

Answer 6

oh wait kinda get it

Answer 7

well the rectangular pieces are relatively easy to find

3*1 = 3 square km

4*1 = 4 square km

5*1 = 5 square km

Answer 8

so we have 3 separate areas of 3, 4, and 5 square km

Answer 9

this is all for the rectangular blue pieces

Answer 10

the green circular pieces are going to be trickier to find, but it's doable

Answer 11

you just need the central angles to find the area of each circular sector

Answer 12

so how do you find those?

Answer 13

is it you find the inside angle of the triangle and the you can find it? Using geometry..

Answer 14

more like trigonometry (since you'll use sine, cosine or tangent)

Answer 15

but yes, you find the inner angles of the triangle first

Answer 16

Ok thanks, i just got mixed up a little! We kept changing math teachers in school and they never really finish each chapters..

Answer 17

I gotcha, so how would you find the inner angles

Answer 18

i got 120 for the inner angle of the sector

Answer 19

the bottom left sector has an angle of 90

so that area is (90/360)*pi*1^2 = pi/4

Answer 20

the top sector has an angle of

360 - 90 - 90 - arctan(3/4) = 143.1301 degrees

so the top sector has an area of (143.1301/360)*pi*1^2 = 0.39758*pi

Answer 21

finally, the bottom sector on the right has an angle of

360 - 90 - 90 - arctan(4/3) = 126.8699 degrees

so the bottom right sector has an area of (126.8699/360)*pi*1^2 = 0.35242*pi

Answer 22

so we have these 3 areas for the 3 rectangular pieces: 3, 4, 5

and these 3 areas for the 3 circular pieces: pi/4, 0.39758*pi, 0.35242*pi

add them up and tell me what you get

Answer 23

14.7

Answer 24

adding up pi/4, 0.39758*pi, 0.35242*pi, you should get

pi/4+ 0.39758*pi+ 0.35242*pi = 3.1415926535898

which is basically pi (more or less)

Answer 25

this is because the circular sectors from each corner combine to make a circle of radius 1, which has an area of pi*r^2 = pi*1^2 = pi

Answer 26

so your total area is

3+4+5+pi = 12 + pi

Answer 27

12+pi is the total area of the shaded exclusion zone

Answer 28

Yes yes, thank you so much!

Answer 29

yw