Another interesting feature of a round Earth is that we can produce a triangle with three 90 degree angles by following the surface of the Earth.
This has been confirmed by tracing airplane routes. I saw the video on Youtube a while back, may have to try and dig it up later.
Iām thinking of logistics and equipment. How would you do the measurements? I mean a tape measure and a skewer work great with an orange, but the earth is a bit less wieldy.
Iād never thought of the method you illustrated so well (nor understood Mitchellās method well at all until you did). There surely must be a height the inverted pyramidal piece on top must have to enable it to have the same volume as the four pieces which include a vertical face. Obviously each piece will have an equal share of frosting ā¦ well, unless angle of the slices failed to give the same amount of frosting to the inverted pyramid around the cut edges as the other four pieces. It seems close enough for government work but seems like a more difficult case to make than Daleās method.
Iām trusting @mitchellmckain for the math. He seems good for it.
Maybe he, like I, has no tolerance for the wailing of disgruntled kids at birthday parties gone sour.
āEven Steven,ā my aunt used always to say, when pouring Vernorās gingerale for my sister and me. You couldnāt have gotten the volumes more precise with lab instruments.
My first thought is: what are the differences between the sides of a Euclidean triangle and the sides of a Reuleaux triangle? Once I have a grasp of the differences, then I look for engineers and technicians to tell me how to measure a distance between two points on a flat earth and how to measure a distance between two points on a spherical earth.
Ferāinstance:
You can imagine how important careful weights and measures would be in ensuring an equal measure of desserts and such in a family of nine (parents included). My wife was one of four children so to this day one of us divides and the other chooses. (Pro tip: when dividing, always lick off the knife or spoon and any tray or container involved.)
My mother had my sister and I do the same thing. Being the oldest, I gamed the system. If we were splitting a soda I would have a wider cup and a narrower cup with ice. I would pour soda into the narrower cup with ice to a level that was just a little higher than the wider cup. My sister thought she was getting one over on me when she chose the cup that had the higher level. We still laugh about it to this day.
6 kids in our house: 4 males, 2 females: Oldest - male; 2nd - male, 3rd male, 4th - female, 5th - female, Youngest - male. Eyeballing āwho got whatā while eating, licking, or drinking. Cakes and āhelping Momā make them, then frost them were the most āprovocativeā events: all 6 eager to lick the mixing bowls and mixing utensils.
I was surprised that the intersection in the interior was actually the middle of the surface. I thought I was going to have to solve simultaneous equations and find an off-center axis to get the five areas equal. (Sorry about my messy finger-drawn sketch ā that may have been misleading since the areas arenāt well represented and proportionate.)
In my home growing up I was second eldest. Before and after me were both boys, then my only sister, then three more boys. In my family male privilege looked like my sister helping my mother in the kitchen daily and helping her keep clean the bathrooms regularly. For we three eldest boys we took turns doing yard work in season and helped with childcare with the younger boys as needed. That my sister was held accountable for the marksmanship of those who stood to pee still makes me cringe. It just is harder to notice privilege in the moment when youāre in the favored class.
āMale privilegeā?? Whatās that? 
Yes. ; - ) Thatās actually good theology, that both are true ā we are both sinners and we both have a God-shaped void, a void that only he can fill.
It certainly isnāt as obvious in the law anymore but we still hold the one left carrying the fetus more responsible for actions out of marriage which are more often pushed for more strongly by the one less encumbered.
Itās @mitchellmckainās. Which cuts to the chase. A face above base gives the same icing. Brilliant. The square based pyramid on a 1 ft cube cake needs to be 7.2" deep, so two diagonal cuts down 4.8" from the inverted base for a start. Slide the knife half way out, keep the point still, rotate down. Now for the angle of the pyramid facesā¦
arctan(2h/l)
That is brilliant - and elegant too; if not as simple to cut as Daleās. Your illustration is perfect to show whatās going on there, Kendel - worth a thousand words! All it takes is to adjust the height of that center point indeed! Then the kiddos can just fight over who wants one of the pieces with frosting on the side, and who gets the pyramid piece with frosting on its base!
As a youngster of five siblings, he who cut the cake had last choice of piece. That way, the slices were always roughly equal to within a Planck mass.
As one of the eldest I always found a dime here or quarter there could secure more from of the youngest. Iām sure if weād been raised in a nest Iād have backed one of them clean out of the tree and the nest just to get their worms. 
He who troubles not to find his tools has little room to complain but something is amiss here since the base of the triangle at the top of your drawing is clearly less than four units.
Just to show that the volume of cake found in each piece Daleās way must be equal. Letās consider that piece and the one immediately to its right.
The top one is a triangle with a base which should be 4 units long and the perpendicular height of the triangle from that base is 2.5 units.
A = (4 * 2ā¦5)/ 2 = 5 sq un
The piece to its right is composed of triangular sections. The bases (a and b) of the two sections must add to 4 and each triangleās perpendicular height to its base is also be 2.5
So A = [(a+b)*2.5]/2 = (4 * 2ā¦5)/ 2 = 5 sq un
I still think Daleās solution is correct, so perhaps I need to prove it one way or another.
It comes down to the areas on the top of the cake since the area of the sides are proportional to the outside perimeters and the volumes are proportional to the areas on top.
The two pieces 4/5 of an edge are easy 1/2 base times height = 1/2 (4/5 a) (1/2 a) = 1/5 a^2.
The symmetrical one with 2/5 of an edge on each side, I cut into three right triangles. One is 2/5 of an edge on each side and the other 2 are (sqrt2/2 a - sqrt2/5 a) on one side and sqrt2/5 a on the other side. So the areas are (1/2)(2/5 a)(2/5 a) = 2/25 a^2 for the one and (sqrt2/2 a - sqrt2/5 a)(sqrt2/5 a) = 3/25 a^2 for the other two together. So the total is 1/5 a^2.
I cut the two remaining pieces into a rectangle 1/5 by 1/2 on a side and two right triangles ā one being 1/2 on a side and (1/2-1/5) on the other, the last triangle being (3/5-1/2) on a side and 1/2 on the other. This gives us 1/10 a^2 + 3/40 a^2 + 1/40 a^2 = 1/5 a^2.
So it all checks out. The areas on the top of the four pieces in Daveās solution are equal and that means the volumes and icing are the same as well.
I think Kendelās analysis is wrong because its is looks to me like she cut the cake into 5 equal angles rather than equal portions of the square perimeter.
But thank you Kendel for taking the time to draw the diagram for my solution ā I am sure that helps people see it better.
Oopsā¦ I guess that math took me too longā¦ Mark beat me to it.
āLet your conversation be always full of grace, seasoned with salt, so that you may know how to answer everyone.ā -Colossians 4:6
This is a place for gracious dialogue about science and faith. Please read our FAQ/Guidelines before posting.