Plane figures: area and perimeter
$A$: area · $P$: perimeter (the sum of all the sides).
Square
$A = c^2$
$P = 4c$
Rectangle
$A = b \cdot h$
$P = 2b + 2h$
Triangle
$A = \dfrac{b \cdot h}{2}$
$P = \text{sum of the 3 sides}$
Rhombus
$A = \dfrac{D \cdot d}{2}$
$P = 4c$
Parallelogram
$A = b \cdot h$
$P = \text{sum of the sides}$
Trapezium
$A = \dfrac{(B + b) \cdot h}{2}$
$P = \text{sum of the sides}$
Regular polygon
$P = n \cdot c$
$A = \dfrac{P \cdot a}{2}$
$n$: number of sides · $a$: apothem · $r$: radius. In a regular hexagon the side equals the radius, $c = r$, and the apothem comes from Pythagoras: $a^2 = r^2 - \left(\tfrac{c}{2}\right)^2$.
Circle
$A = \pi r^2$
$L = 2\pi r$
$L$: length of the circumference
Circular sector
$A = \dfrac{\pi r^2 \cdot \alpha}{360^\circ}$
$L_{\text{arc}} = \dfrac{2\pi r \cdot \alpha}{360^\circ}$
$P = L_{\text{arc}} + 2r$
Annulus
$A = \pi \left(R^2 - r^2\right)$
$P = 2\pi R + 2\pi r$
Pythagoras' theorem
$a^2 = b^2 + c^2$
In a right triangle, $a$ is the hypotenuse.
Solids: area and volume
$A_B$: base area · $A_L$: lateral area · $A_T$: total area · $P_B$: perimeter of the base · $V$: volume.
Prism
$A_B = \text{area of the base polygon}$
$A_L = P_B \cdot h$
$A_T = 2A_B + A_L$
$V = A_B \cdot h$
Cube with edge $c$: $A_T = 6c^2$, $V = c^3$.
Pyramid
$A_B = \text{area of the base polygon}$
$A_L = n \cdot A_{\triangle}$
$A_L = \dfrac{P_B \cdot a_p}{2}$
$A_T = A_B + A_L$
$V = \dfrac{A_B \cdot h}{3}$
The side faces are $n$ equal triangles. 1st way: $A_L = n \cdot A_{\triangle}$, with $A_{\triangle} = \tfrac{c \cdot a_p}{2}$ (base $c$, side of the polygon; height $a_p$, slant height). 2nd way: since $n \cdot c = P_B$, altogether $A_L = \tfrac{P_B \cdot a_p}{2}$. If the base is a regular polygon, $A_B = \tfrac{P_B \cdot a_b}{2}$ ($a_b$: apothem of the base). Pythagoras: $a_p^2 = h^2 + a_b^2$.
Cylinder
$A_B = \pi r^2$
$A_L = 2\pi r \cdot h$
$A_T = 2A_B + A_L$
$V = A_B \cdot h = \pi r^2 h$
Cone
$A_B = \pi r^2$
$A_L = \pi r \cdot g$
$A_T = A_B + A_L$
$V = \dfrac{A_B \cdot h}{3} = \dfrac{\pi r^2 h}{3}$
$g$: slant height, $g^2 = h^2 + r^2$.
Sphere
$A = 4\pi r^2$
$V = \dfrac{4}{3}\pi r^3$
No base: just the total area. Hemisphere: $V = \tfrac{2}{3}\pi r^3$.
Capacity and units
$1\ \text{dm}^3 = 1\ \text{L}$
$1\ \text{m}^3 = 1000\ \text{dm}^3 = 1000\ \text{L}$
$1\ \text{cm}^3 = 1\ \text{mL}$
Areas: each unit step is $\times 100$. Volumes: each step is $\times 1000$.
Notice the difference: prisms and cylinders have two bases ($A_T = 2A_B + A_L$); pyramids and cones have only one ($A_T = A_B + A_L$), and their volume is a third of that of the prism or cylinder with the same base and height.