# Help on working out the area of this property!

• May 29th 2012, 04:59 AM
ishaun707
Help on working out the area of this property!
I am having trouble working out this question so please be detailed as possible :)

Crawford's Uncle Bob has a very large property with an unusual pentagonal shaped boundary.
Crawford is asked to determine its area in square kilometers.
He visits the Lands Department and obtains the following sketch from survey records.

Show how Crawford calculated the area of the property.

Attachment 23989

Thanks!
• May 29th 2012, 06:15 AM
emakarov
Re: Help on working out the area of this property!
Quote:

Originally Posted by ishaun707
I am having trouble working out this question so please be detailed as possible :)

I am not going to show all calculations, but feel free to ask questions and post your work to be checked.

Quote:

Originally Posted by ishaun707
Show how Crawford calculated the area of the property.

I don't think we can tell for sure how Crawford did it (Smile) Maybe he used Monte Carlo method.

Attachment 23990

You can find AC and AD using the law of cosines. Then you can find angles BAC and DAE using the law of sines. This way, you can find angle CAD. Then you can find the areas of triangles BAC, CAD and DAE using cross product.
• May 29th 2012, 06:26 AM
ishaun707
Re: Help on working out the area of this property!
Sorry to ask again but I'm in grade 9 in Australia and we haven't learnt sine or cosine laws yet
so I don't think the teachers expect us to do that. I heard from others that you split the shape into
other shapes so you can work it out and add them up together. Could you please show me that way please?
• May 29th 2012, 06:27 AM
emakarov
Re: Help on working out the area of this property!
Which facts that you learned may be relevant to this problem, do you think?
• May 29th 2012, 06:34 AM
ishaun707
Re: Help on working out the area of this property!
Areas of rectangle, triangle, trapezium, pythagoras.
Just basic skills. One thing I know is that you split into multiple shapes to add them up at the end.
Oh and also, the previous question mentioned something with accurate scale diagram but I'm not sure if that applies to this :S
Are the angles relevant too?
• May 29th 2012, 06:52 AM
emakarov
Re: Help on working out the area of this property!
Quote:

Originally Posted by ishaun707
Areas of rectangle, triangle, trapezium, pythagoras.

But I assume you have learned sines and cosines, right? Then it is easy to find the coordinates of all vertices. Let A have the coordinates $(x_1, y_1)$, ..., E have the coordinates $(x_5,y_5)$ (we can assume $x_1=y_1=0$). Then according to the surveyor's formula, the area is ${1 \over 2}|x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - x_2y_1 - x_3y_2 - x_4y_3 - x_5y_4 - x_1y_5|$.

I'll think of other ways.
• May 29th 2012, 06:54 AM
ishaun707
Re: Help on working out the area of this property!
We haven't learnt any sine or cosines or surveyor's formula.
I think its just basic triangle and rectangle shapes but I don't know to cut the shape so I can add it up at the end.
• May 29th 2012, 07:50 AM
BobP
Re: Help on working out the area of this property!
Apologies for the interuption, but isn't the area insufficiently defined for an area to be calculated ?
Using the notation from the second post of this thread, can't the length CD be increased (increasing the area) or decreased (decreasing the area) without altering the other lengths or given angles ? Unless, for example, the angle BAE is being assumed to be a rectangle ?
• May 29th 2012, 07:59 AM
emakarov
Re: Help on working out the area of this property!
Quote:

Originally Posted by BobP
Using the notation from the second post of this thread, can't the length CD be increased (increasing the area) or decreased (decreasing the area) without altering the other lengths or given angles ? Unless, for example, the angle BAE is being assumed to be a rectangle ?

I assumed that angle BAE is 90 degrees.

I don't see how to calculate the area precisely without trigonometric functions.

Quote:

Originally Posted by ishaun707
Oh and also, the previous question mentioned something with accurate scale diagram but I'm not sure if that applies to this :S

Maybe the idea is to draw an accurate scale diagram of the shape on the graph paper and count the number of rectangles inside the diagram.
• May 30th 2012, 03:54 AM
bjhopper
Re: Help on working out the area of this property!
Using post 2 lettersand assuming angle A is 90 deg I calculate that the area is 1164 km^2.Horizontal and vertical lines are drawn thru B,C,D and F creating 3 right triangles and 2 rectangles.I use the simple sin,cos and tangent to id all angles and lines
• May 30th 2012, 04:05 AM
emakarov
Re: Help on working out the area of this property!
Hmm, I am not sure about 3 right triangles and 2 rectangles.

Attachment 23996
• May 30th 2012, 05:22 AM
bjhopper
Re: Help on working out the area of this property!
Hello emakarov,
Your new drawing is like mine but two line segments must be erased.