+Euclidean

See \cite{hallStevens}. Deals with $R^k$, the Euclidian $k$ space, which is described in the vector spaces survey.

Considers metric properties such as distances between points. Perpendiculars, parallels, projections.

Relationship between angles at the intersection of a line with parallels or in a triangle (and other polygons). Similarity and congruence of triangles (and other polygons). Angle bisectors, Medians in a triangle intersect at incenter, circumcenter.

Circle; lines, angles, triangles, chords, arcs in circles. Tangents, Intersection of circles. Ellipsoids, spheres.

Polytopes: an object in $R^n$ whose boundary surfaces are flat. Convex polytopes: polytopes which are also convex sets.

Area/ volume

The notion of area/ volume in case of euclidean spaces corresponds to the box (Lebesgue) measure over the euclidean space. This is described in the vector spaces survey.

Ellipse, circle. Surface area of n-ball: $\frac{d{V}_n}{dr}$.

The r-radius $n-1$ hypersphere $S^{n-1}=\{x\in R^n:\parallel x\parallel =r\}$: a n-1 dim manifold. Encloses a n-ball with volume $V_n=\frac{{\pi }^{\frac{n}{2}}{r}^n}{\mathrm{\Gamma }(\frac{n}{2}+1)}$ \why.

Trigonometry

Obvious with the right construction: sin (A+B) = sin A cos B + cos A sin B. cos (A+B) = cos A cos B - sin A sin B. Trigonometry in triangle calculations. ${\sin }^{-1}x,{\cos }^{-1}x$.