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1/16/2020 9:48:49 PM
Posted: 11/19/2012 4:03:00 PM EST
[Last Edit: 11/19/2012 5:14:33 PM EST by setlab]
I'm building a gas forge for blacksmithing/bladesmithing work and I want to know the inside volume so I can figure out how many burners I need. The inside of the forge I'm copying looks something like this picture, a circle with a section cut off the bottom. What is the formula I need to use to find the area?

Edit: these pictures should work.

Link Posted: 11/19/2012 4:56:07 PM EST
Your picture is missing.
Link Posted: 11/19/2012 4:56:08 PM EST
Google "VOLUME OF A CYLINDER" I cant see your PIC...I assume that's what you're looking for.
Link Posted: 11/19/2012 5:09:24 PM EST
[Last Edit: 11/19/2012 5:12:49 PM EST by setlab]
I'll look for another picture that works here. Imagine a inverted "D" shape basically I have a 9.75" inner forge diameter that I will be adding a firebrick floor to. So far I'm not 100% positive on the height I want to make the floor.

Link Posted: 11/19/2012 7:01:19 PM EST
[Last Edit: 11/19/2012 7:02:22 PM EST by Chapman]
Just want the inner (open) volume?


Where pi = 3.14159
D^2 = Diameter of opening, squared
h = depth of the forge

I know it seems elementary, but make sure you use consistent units. Don't measure the diameter in inches, then the depth in feet. Your calculations won't come out right

If your burner specifications call for a certain number of cubic feet, just make sure to measure the diameter and depth in feet. The resulting volume will be in cubic feet.

ETA: Your actual volume will be a bit less than this, due to the floor. The formula given is for the volume of a cylinder. It shouldn't throw you off by too much, however.

Link Posted: 11/19/2012 7:14:56 PM EST
Link Posted: 11/19/2012 7:35:30 PM EST
Originally Posted By 230JHP:

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Link Posted: 11/19/2012 7:56:18 PM EST
Thanks guys, but neither of those formulas work. Both of those formulas just give the area of a circle/cylinder, but I did learn that R^2*pie can be re-written in more then one way!

Using R^2*pie the area without the firebrick is 74.66191 square inches.
The fire brick area is 27 square inches.

I can rough estimate from that and say the area with the fire brick will be more than 47.66191 square inches by maybe +5 square inches. and I guess that is a good enough estimate. I could get a better estimate by subtracting the corners off the firebrick by using the formula for the area of a triangle too. But there has to be a way to calculate the the area of a circle cut by a plane that isn't in the center.

I'm going to be using 3/4" burners and in order to achieve reliable welding temperatures I will need 1 burner per 300-350 cubic inches.
Link Posted: 11/19/2012 10:43:23 PM EST
The issue is that a formula would have to be derived, mathematically, for your particular circumstance. If you give me the circular diameter of the opening, along with the smallest diameter, I can plug it in to a NX and get you an area.

What I mean is measure between 3 o clock and 9 o clock, for the circular diameter. Then measure between 12 and 6 to get the smallest diameter of the furnace. With that information, I may be able to get you an exact number.

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Link Posted: 11/20/2012 5:21:31 AM EST
Ah, I see what you need now. What you need to do is find the area of the circle subtended by a segment. First measure the board width at the points it intersects the circle. We'll call that "C".

Now calculate the central angle, theta , using trigonometry and knowing the radius (half the measured diameter). The triangle is best understood split into two right triangles (i.e. drop a line straight down the middle from the center to line C) to calculate the central angle. This is governed by the equation:

sin(theta/2) = (c/2)/R

Therefore: theta = 2*arcsin[c/(2*R)]

Once we know that, we can calculate the area of the circular segment using this forumla:

A_segment = (R^2)/2 * [theta*pi/180 - sin(theta*pi/180)]

So if your chord length, C, is 10 inches, your radius, R, is 15 inches: theta = 38.94° and A_segment = 75.12 sq. in.

We can then subtract that area from the total circular area to find the open area. The circular area is pi*R^2.

If you want to know the volume, you multiply the open area by the depth.

Now if you want to know the surface area inside the furnace, that's different but not difficult. Just let me know if that's what you're actually looking for.

Link Posted: 11/20/2012 10:06:18 AM EST

the mising piece at the bottom is called a segment of a circle.

Compute its area,.
Look here.

Subtract that are from the circle pi * r^2

Now multiply by the depth of the furnace front to back.
Link Posted: 11/20/2012 1:59:46 PM EST
[Last Edit: 11/20/2012 2:30:07 PM EST by setlab]
Awesome, got the area all worked out. Now it's time to build. Thanks guys.

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