Geometry/Appendix A

This is an incomplete list of formulas used in geometry.

Polygon

 * Sum the lengths of the sides.

Circle

 * $$\pi d\ = 2\pi r\,$$
 * $$d\,$$ is the diameter
 * $$r\,$$ is the radius

Triangles

 * Law of Sines: $$\frac{a}{sin(A)}=\frac{b}{sin(B)}=\frac{c}{sin(C)}$$
 * $$a, b, c\,$$ are sides, $$A, B, C\,$$ are the angles corresponding to $$a, b, c\,$$ respectively.
 * Law of Cosines: $$c^2 = a^2 + b^2 - 2ab\cos(C),$$
 * $$a, b, c\,$$ are sides, $$A, B, C\,$$ are the angles corresponding to $$a, b, c\,$$ respectively.

Right Triangles

 * Pythagorean Theorem: $$c^2=a^2+b^2$$
 * $$a, b, c\,$$ are sides where c is greater than other two.

Triangles

 * $$A=\frac{bh}{2}\,$$
 * $$b\,$$ = base, $$h\,$$ = height (perpendicular to base), $$A\,$$ = area
 * Heron's Formula: $$A=\sqrt{s(s-a)(s-b)(s-c)}\,$$
 * $$a, b, c\,$$ are sides, and $$s = \frac{a+b+c}{2} \,$$, $$A\,$$ = area

Equilateral Triangles

 * $$\frac{\sqrt{3}a^2}{4}\,$$
 * $$a\,$$ is a side

Squares

 * $$s^2\,$$
 * $$s\,$$ is the length of the square's side

Rectangles

 * $$ab\,$$
 * $$a\,$$ and $$b\,$$ are the sides of the rectangle

Parallelograms

 * $$bh\,$$
 * $$b\,$$ is the base, $$h\,$$ is the height

Trapezoids

 * $$\frac{(b_1+b_2)h}{2}\,$$
 * $$b_1,b_2\,$$ are the two bases, $$h\,$$ is the height

Circles

 * $$\pi r^2\,$$
 * $$r\,$$ is the radius

Surface Areas

 * Cube:   6×($$s^2$$)
 * $$s\,$$ is the length of a side.
 * Rectangular Prism: 2×(($$l,$$ × $$w\,$$) + ($$l\,$$ × $$h\,$$) + ($$w\,$$ × $$h\,$$))
 * $$l\,$$, $$w\,$$, and $$h\,$$ are the length, width, and height of the prism
 * Sphere:  4×π×($$r\,$$2)
 * $$r\,$$ is the radius of the sphere
 * Cylinder: 2&times;π&times;$$r\,$$&times;($$h\,$$ + $$r\,$$)
 * $$r\,$$ is the radius of the circular base, and $$h\,$$ is the height
 * Pyramid: $$A = A_b + \frac{ps}{2}$$
 * $$A$$ = Surface area, $$A_b$$ = Area of the Base, $$p$$ = Perimeter of the base, $$s$$ = slant height.
 * The surface area of a regular pyramid can also be determined based only on the number of sides($$n$$), the radius($$r$$) or side length($$l$$), and the height($$h$$)
 * If $$r$$ is known, $$l$$ is defined as $$l = \sqrt{(rcos(\frac{360}{n})-r)^2 + (rsin(\frac{360}{n}))^2} = \sqrt{2}r\sqrt{1-cos(\frac{360}{n})}$$
 * or if $$l$$ is known, $$r$$ is defined as $$r = \frac{l}{\sqrt{2}\sqrt{1-cos(\frac{360}{n})}}$$
 * The slant height $$h_1$$ is given by $$\sqrt{r^2+h^2+\frac{l^2}{4}}$$
 * The total surface area of the pyramid is given by $$n\frac{l}{2}[h_1 + h_0]$$


 * Cone: π&times;r&times;(r + √(r2 + h2))
 * $$r\,$$ is the radius of the circular base, and $$h\,$$ is the height.

Volume

 * Cube $$s^3 = s \cdot s \cdot s$$
 * s = length of a side
 * Rectangular Prism $$l \cdot w \cdot h$$
 * l = length, w = width, h = height
 * Cylinder(Circular Prism)$$\pi r^2 \cdot h$$
 * r = radius of circular face, h = distance between faces
 * Any prism that has a constant cross sectional area along the height:
 * $$A \cdot h$$
 * A = area of the base, h = height
 * Sphere: $$\frac{4}{3} \pi r^3$$
 * r = radius of sphere
 * Ellipsoid: $$\frac{4}{3} \pi abc$$
 * a, b, c = semi-axes of ellipsoid
 * Pyramid: $$\frac{1}{3} A h$$
 * A = area of base, h = height from base to apex
 * Cone (circular-based pyramid):$$\frac{1}{3} \pi r^2 h$$
 * r = radius of circle at base, h = distance from base to tip