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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.

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Thanks!

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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.

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?

Re: Help on working out the area of this property!

Which facts that you learned may be relevant to this problem, do you think?

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?

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 $\displaystyle (x_1, y_1)$, ..., E have the coordinates $\displaystyle (x_5,y_5)$ (we can assume $\displaystyle x_1=y_1=0$). Then according to the surveyor's formula, the area is $\displaystyle {1 \over 2}|x_1y_2 + x_2y_3 + x_3y_4 + x_4y_5 + x_5y_1 - x_2y_1 - x_3y_2 - x_4y_3 - x_5y_4 - x_1y_5|$.

I'll think of other ways.

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.

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 ?

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.

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

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Re: Help on working out the area of this property!

Hmm, I am not sure about 3 right triangles and 2 rectangles.

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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.