I need to solve it by using Geometry equations and it has been a long time since I have seen Geometry. I need the cross sectional area, volume of the dovetail slide, and the surface are of the dovetail slide.

I could really use your help.

(C-Is long side of right triangle. This is oposite the right angle) Squared = (A-label one of the other sides of the triangle) Squared + (B-label one of the other sides of the triangle) Squared

Volume of a right angled body with straight sides is Side * Side * Height. This will give you the total volume you cannot go over. 2 * 3.5 * 5. Find the volume of the dove tail and then find the volume of the body itself separately. The should total 2 * 3.5 * 5.

Surface area is length * width only if it is four straight lines and four right angles.

Don't worry about the problem, rebuild your understanding of volume, surface area and the rest. Make it like water and you will enjoy it. If not, what is the real purpose of doing?

Secondly the volume of an object with two parallel bases, and a constant cross section (pardon me for the terms) is Base*Height

It is right that area surface is length*width only for rectangular shapes, but what i did is that i mapped the surface of the dove-tail with a rectangle, and found the surface of the rectangle, which will be equal to the surface of the dove tail (After adding the surface of the two bases).

Also you cannot calculate the volume by simnply saying side*side*height, cause the base is not rectangular, it has a special shape. (as you said this will be a calculation of the maximum)

It is like if you have a tube, can you simply say side*side*length ? sure not, you will have to calculate the surface of the base and then multiply it by the height.

As to the volume, that calculation is for the volume of the entire object without the cut(this was specified). One who has been out of the process for a while could take the formulas given compute the total area. The the area of the right triangles which form the in cut. Add this to the rectangular cut and you have the area of the cutout she could then reapply the routine to get the physical object itself add them together and they should equal the total. It is a good exercise to get back into it.

The wise comments were intended for Alex not you, but if you feel the need partake.

Teach how to fish instead of just giving them just a fish. That is of course unless they can mulitple fish.

Thanks
lena2383

First of all i will redraw the drawings of lena for the sake of clarity.
One thing is not really clear in the drawing of lena, which is wether the depth is 5 cm everywhere, so i will assume this (and clarify it in the drawing).
We will also agree to call the heading face in the drawing as the base.

drawing 1

1-Calculating the area of the base (=the area of the cross section)
A geometric way of calculating the area of a non-simple shape is to make that shape into smaller shapes (That make up the original shape), and then calculate the area of each little shape alone, and then add the areas of all little shapes, you will end up with the area of the big shape (the base in our example).
You can partition the big shape in lot of ways, i have chosen one of them.

Drawing 2

As you can see i made the base into 5 parts.
Triangles (in red and purple)
Rectangles (in yellow, blue and brown)

Now if we find the areas of all those little parts, and add them, we will end up with the area of the base itself !

We know that the area of a rectangle = the first side * the second side
and the area of a triangle = 0.5*the base line*the height
(remember that base side is any side of the trangle, and the height is a line perpendicular on the base (that we chose) and ending at a point of a triangle)

now the surface of the cross section (which equals the surface of the base) (call it B) would be :
B=A_red+A_yellow+A_blue+A_brown+A_purple (where A_red stands for the area of the red part, and so for the other variables)

Now note that that the dove tail's base is symetric, so we can simplify our last equation (please note that you don't have to do this step, but doing it will make it easier)

And because of symetry, the area of the red part equals the area of the purple part, and the area of the brwon part equals the area of the yellow part.

Drawing 3

So we can rewrite the value of B as:
B=(2*A_red)+(2*A_yellow)+A_blue

Great! Now only find the area of the red and yellow and blue parts and we are finished of the the area of the base !

(please note that the colors of the line in the following drawing has nothing to do with the colors of the parts)

Drawing 4

Now you can notice that :
A_red=0.5*(The Brown Line)*(The Purple Line)
A_Yellow=(The Green Line)*(The Yellow Line)
A_blue=(The Red Line)*(The Blue Line)
(in the future and for clarity, we will write instead of (The Brown Line) L_brown, and same for the other lines)
A_red=0.5*L_Brown*L_Purple
A_Yellow=L_Green*L_Yellow
A_blue=L_Red*L_Blue

Some of the values of the lines' length are known, we will have to find out some other values.

(From the original drawing (drawing 1))
L_Green=2.0 cm
L_Brown=0.4 cm
L_Red=2.2 cm

And for the other lines
L_Blue=2.0 cm - 0.4 cm = 1.6 cm
Note that the yellow line and the red line, and another line with the same length of the yellow line add up to 3.5 cm
So the length of the twice the yellow line (2*L_yellow) = 3.5-2.2=1.3
And therefore L_Yellow=1.3/2 = 0.65 cm
Also note that the purple line and the line parallel to the yellow line are equal to the light blue line
So L_{Light Blue}=L_Purple+L_Yellow
But also note that the two light blue line and the distance between them add up to 3.5
2*L_{Light Blue}+1.9=3.5
2*L_{Light Blue}=1.6
L_{Light Blue}=0.8 cm
L_Yellow+L_Purple=0.8
But L_yellow=0.65
Therefore L_purple+0.65=0.8
So L_Purple=0.15 cm

Now we have the values of the length of all the lines that we need, we can plug them back into the equations of area, and find the areas of the little shapes

A_red=0.5*L_Brown*L_Purple=0.5*0.4*0.15=0.03 cm²
A_Yellow=L_Green*L_Yellow=2.0*0.65=1.3 cm²
A_blue=L_Red*L_Blue=2.2*1.6=3.52 cm²

Plug them back into the equation of B
B=(2*A_red)+(2*A_yellow)+A[sub]blue[/sub)
B=(2*0.03)+(2*1.3)+3.52
B=0.06+2.6+3.52=6.18 cm²

So we found the area of the base !

(coming up: the area of the surface of the dove tail, and the volume of the dove tail)
Lena, tell me if that is what you were looking for

Drawing 5

So applying this rule, and since we know B (we calculated it in the first part of the question), and we know h to be 5.0 cm (from the givings in the original drawing)
v=B*h=6.18*5.0=30.9 cm³

And this ends part 2 of the question

3-The area of teh surface of the dove-tail
The easier way to calculate the area of the surface of a body (this is my personal opinion that this is the easier way) is to imagine the body like if it made from a paper, and fold out the paper, and calculate the area of the paper.

I made this for the dove tail, and reached the following:

Drawing 6

The colors in this drawing has nothing to do with previous colors.
When the paper is folded back, each two lines that have the same color will meet together.
Now the area of this paper equals B+B+(the area of the rectangle in the middle
If i call the area of the surface of the paper (which is equal to the area of the surface of the dove-tail) as S, and i call the area of the rectangle in the middle as X then:
S=2B+X
Now B is known for us, all we need to know is X
X=(first side)*(second side)
From the drawing you can see that one of the sides of the rectangle is 5.0 cm long, and the other one is the sum of 0.8+2.0+2.2+0.8+2.0+3.5+(the little blue line)+(the little purple line).
You can find the length of the little blue line and the little purple line from the phythagorean theorom (sorry for spelling ).

Drawing 7

(The value of B is from the original drawing (Drawing 1), and the value of A is what we found under Drawing 4 (the purple line in Drawing 4))
According to the theorom:
C²=A²+B²
C²=(0.15)²+(0.4)²
C²=0.0225+0.16
C²=0.1825
Therefore :
C=~0.4272 cm

Now note that C is the little purple line in Drawing 6, and is equal to the little blue line in Drawing 6 (for symetry).

Now back to the side of the rectangle in Drawing 6.
The side=~0.8+2.0+0.4272+2.2+0.4272+0.8+2.0+3.5=~12.1554 cm

Back to the fact that the area of the rectangle is equal to its first side times its second side
X=12.1554*5=60.777 cm²
And
S=2B+X=(2*6.18)+60.777=12.36+60.777=73.137 cm²

That's all
I hope i helped.(and didn't make too much mistakes)