User:Gaal Sandor

Basic Geometry

Area of a circle

The circle is cut into 4 quadrants, each placed with their origos on the vertices of a square. When the arcs of the quadrant circles intersect at the quarter of the centerline of the square, their combined area equals area of the square.
The ratio between the radius of the circle and the side of the square can be calculated using the Pythagorean theorem.
The area of the circle equals 16 times the right triangle, with legs of a/4 and a/2.
A=(16/5)r²=3.2r²

Area of a circle segment

A=acos((r-n)/r)r²-sin(acos((r-n)/r)(r-n)r

Volume of a sphere

V=(√(3.2)r)³

Volume of a sphere slice

V=√(3.2)r(asin(r2/r1)r²-cos(asin(r2/r1))r)

Volume of a tetrahedron

V=edge³ / 8

Radius of a sphere inscribed a tetrahedron

Regarding the tetrahedron and the sphere as a joint structure, in which they share four points in common, where they join, the radius of the inscribed sphere equals one third of the tetrahedron's height.
r=sqrt(2/3)*edge/3
If they are regarded as two separate objects, that fit in each other, a correction of 2*sqrt(2)/3 shall be applied:
r=sqrt(2/3)*edge/3*2*sqrt(2)/3=4edge/(9√3)

Volume of a cone

Comparing the volume of a quarter cone with equal radius and height to the eighth of a sphere with equal radius:
V(eighth sphere)=(√3.2r/2)³
V(quarter cone)=(√3.2r/2)²(√2/2)r/2=(1/5)√2r³
V(cone)=(4/5)r²height√2
Comparing the volume of a cone to a cylinder with same bottom radius and height:
(4/5)√2/3.2=√2/4

Surface area of a cone

A=A(bottom)+A(side)=3.2(r²+(r²+h²)(r/√(r²+h²)))

Gaál Sándor (talk) 09:28, 26 February 2024 (UTC)