Oktay Mercimek, EMAT 6690

**Estimating
Volume of an Irregular Solid**

This write up is a additional exploration for Volume of An Irregular Solid project and I recommend you to read that project first. there are a lot of way to find the volume of regular shapes but there is no certain way to compute the volume of irregular shapes. In this write up I want to use little different way find the volume of waste. The difference will be the computation. Computation will be made by Geometer's Sketchpad in a completely automated way and the main idea is keeping the process as simple as possible.

**SECOND
APPROACH FOR AVERAGE VALUE**

**(Probably
this approach is much more appropriate for high school students. That's why I
put my first approach to end of this page)**

Now I want to make another argument about what can be average amount of volume. I will use only Geometer's Sketch Pad 's most basic functions.

1.First I will define an polygon interior for three cross- section and I will find area using GSP

a. copy pictures and paste them into a gsp file.

b. then draw lines around cross-section (use only one direction, e.g. clockwise)

(half progress)

Tip: In GSP you don”t need to draw lines to define an polygon interior. We draw lines to see how well points cover the picture.

After choosing lines , select only points ( one direction again to select points and this warning is much more important in this step)

d. select “Polygon Interior” under “Construct” menu and then “Area” under “Measure” menu.

Area result may vary related with what kind of GSP settings you use

e. repeat same steps from a to e for BB' and CC'

2. Let's find area factor for each one

a. Draw AA' line

b.
Draw lines perpendicular to AA' at points A and A' ( select line AA'
, point A and point A' then select “perpendicular lines”
under “construct” menu) and define point K that is
intersection of one of the perpendicular lines and 20' feet line.

then
draw a line from point K that is parallel to line AA' and define
point M

now
we can construct rectangular interior for rectangle AA'KM and find
area.

We
know real area of rectangle AA'MK is 20*180=3600 square foot so the
ratio between real area and drawing's are is 3600/12,76=282,09 will remain fixed for
every sample area in this picture. GSP found area of P_{1} as
6,55 cm^{2}. To find (area of P_{1}),

and
(RealAreaP_{1})=282,09*6,55=1848,58

apply same system for BB' and CC'

If we assume each cross-section is in the center of one third of the pile

then
length for each section is 150 ft.

so Volume of Section 1 is 150*1848,58=277287 cubic feet,

Volume of Section 2 is 150*2851,52=427728

and Volume of Section 3 is 150*2332,29=349843,5. total volume is 39068,83 cubic yards. A GSP file that includes all process

**My first and
worse approach (engineers may like it though! )**

I first thought to share the whole region into 5 parts.

← **Image1**

These parts are:

Part between R

_{0}and R_{1}Part between R

_{1}and R_{2}Part between R

_{2}and R_{3}Part between R

_{3}and R_{4}Part between R

_{4}and R_{5}

First
I defined appropriate points on three cross-section.

To see whether this selection true or not, I drew segments.

Next drawings demonstrate my ideas to find volume.

drawing1: this is my system to approximate first (and last) part. I thought sides as pyramids and middle as prisms. In the middle area, I took average height to find heights of prisms so it made my job easier!

You can see average
issue below on points between B_{1} and C_{1} .
Actually I didn't draw all lines on all drawings to show you a more
clear picture.

drawing2:
This is my system to find volume Part 2 between R_{1} and R_{2}
. I used AA' cross-section to find heights of this section. Part 4 is
also similar, only difference is I used naturally CC' cross-section.

drawing
3: and this is for part 3, Part between R_{2} and R_{3}.
BB' cross-section is in the middle and I used heights on BB'
cross-section.

Then I started with
part 1, between R_{0} and R_{1} . The problem was how
to transfer these points to big picture, and I decided to use
similarity of triangles.

The picture below shows how I used similarity.

I drew a circle centered A with radius AA'( red segment, the one in the image 1)

I drew perpendicular lines to AA'.

then assigned the intersection points.

In this picture I used parallel lines to compare AA' on the cross-section picture an AA' on the image 1(big picture).

Then I transferred this lengths to image 1 using circles. (I did same things also for BB' and CC' )

5. I used XY segment to find vertical length on image1. We know XY actually refers to 450 ft.

6. I treated as a
pyramid to section φ_{1}
and φ_{10} , and as a prism to section φ_{2}
through φ_{9} .

For example,

and since we know XY and KM we can find easily what is the real value of KM.

GSP can also do this computation.

I used almost exactly same system to find heights along AA' :

we
can set same equation here

and
we can apply same equation just simply changing O_{1}P_{1}
by zW, ZY, ..., and M_{1}N_{1}.

and
we also need more measurement

we
would be ready to go further when we measure vertical length like
DE,FG, ... ,UV

We did all necessary measurement to find volume of first part so we can compute volume.

so
V_{1}=2644,81 cubic yards

V_{2}=3941,06
cubic yards

Part3:

V_{3}=
11819,85 cubic yards.

Part 4 is similar to Part 2, and

V_{4
}= 5037,32 cubic yards.

Part 5 is similar to Part1, and

V_{5}
=2785,16

Then total volume approximation is 26228. I tried to be as near as possible to real volume amount.