Angle excess and defect
On a sphere the angles of a triangle add up to more than 180°, on the hyperbolic plane to less. Find the area of a triangle or a polygon from its angles.
How to solve it
On the plane a triangle's angles always add up to $180^\circ$. On a sphere they add up to more, and the excess is the triangle's area divided by $R^2$. On the hyperbolic plane they add up to less, and the defect gives the area. The bigger the figure, the stronger the difference from the plane.
Step by step
- Add the angles and convert the sum to radians ($180^\circ = \pi$).
- A sphere: the excess $\varepsilon = (\alpha + \beta + \gamma) - \pi$. For an $n$-gon, the angle sum minus $(n - 2)\pi$.
- The hyperbolic plane: the defect $\delta = \pi - (\alpha + \beta + \gamma)$; for an $n$-gon, $(n - 2)\pi$ minus the angle sum.
- The area: $S = R^2 \cdot \varepsilon$ on a sphere and $S = R^2 \cdot \delta$ on the hyperbolic plane of curvature $-\frac{1}{R^2}$.
Common mistakes
- Using the excess in degrees instead of radians.
- Subtracting $\pi$ for a polygon instead of $(n - 2)\pi$.
- Getting the sign wrong: on the hyperbolic plane the excess is negative, and the area is the defect, positive.
- Forgetting to multiply by $R^2$.