
# 20.9: Area in the neutral planes and spheres


Area can be defined in the neutral planes and spheres. In the definition, the solid unit square $$\mathcal{K}_1$$ has to be exchanged to a fixed nondegenerate polygonal set $$\mathcal{U}$$. One has to make such change for good reason — hyperbolic plane and sphere have no squares.

The set $$\mathcal{U}$$ in this case plays role of the unit measure for the area and changing $$\mathcal{U}$$ will require conversion of area units.

According to the standard convention, the set $$\mathcal{U}$$ is taken so that on small scales area behaves like area in the Euclidean plane. Say, if $$\mathcal{K}_a$$ denotes the solid quadrangle $$\blacksquare ABCD$$ with right angles at $$A$$, $$B$$, and $$C$$ such that $$AB=BC=a$$, then we may assume that

$$\tfrac{1}{a^2}\cdot\text{area }\mathcal{K}_a\to 1 \quad \text{as} \quad a\to 0.$$

This convention works equally well for spheres and neutral planes, including the Euclidean plane. In spherical geometry equivalently we may assume that if $$r$$ is the radius of the sphere, then the area of whole sphere is $$4\cdot\pi\cdot r^2$$.

Recall that defect of triangle $$\triangle ABC$$ is defined as

$$\text{defect}(\triangle ABC) := \pi-|\measuredangle ABC|-|\measuredangle BCA|-|\measuredangle CAB|.$$

It turns out that any neutral plane or sphere there is a real number $$k$$ such that

$k\cdot\text{area }(\blacktriangle ABC)+\text{defect}(\triangle ABC)=0$

for any $$\triangle ABC$$.

This number $$k$$ is called curvature; $$k=0$$ for the Euclidean plane, $$k=-1$$ for the h-plane, $$k=1$$ for the unit sphere, and $$k=\tfrac1{r^2}$$ for the sphere of radius $$r$$.

Since the angles of ideal triangle vanish, any ideal triangle in h-plane has area $$\pi$$. Similarly in the unit sphere the area of equilateral triangle with right angles has area $$\tfrac\pi2$$; since whole sphere can be subdivided in eight such triangles, we get that the area of unit sphere is $$4\cdot\pi$$.

The identity 20.9.1 can be used as an alternative way to introduce area function; it works on spheres and all neutral planes, except for the Euclidean plane.