Midpoint, volume, surface area

Midpoint

The midpoint of two points $\left(x_1, y_1, z_1\right)$ and $\left(x_2, y_2, z_2\right)$ in 3-D space is

\text{midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)

HL students should recognize this as the mean of two position vectors.

shape	area	notes
circle	$\pi r^2$
triangle	$\displaystyle \frac 12 bh$	Height (altitude) may be outside the base, for an obtuse triangle
parallelogram	$bh$
rectangle	$lw$
trapezoid (trapezium)	$\displaystyle {\frac 12(a+b)h}$	$a$ and $b$ are the two parallel bases

Circumference of a circle: $2\pi r$

Perimeter of a square: $4l$

Perimeter of a rectangle: $2(l+w) = 2l + 2w$

3D shape	volume	surface area	notes
sphere	$\displaystyle \frac 43\pi r^3$	$4\pi r^2$
semisphere	$\displaystyle \frac 23\pi r^3$	$3\pi r^2$	including base
prism	$Ah$		$A$ : base area
pyramid	$\displaystyle \frac 13 Ah$		$A$ : base area
cylinder	$\pi r^2h$	$2\pi r(r + h)$	$2\pi rh$ is for the curved surface
cone	$\displaystyle \frac 13 \pi r^2h$	$\pi r (r + l)$	$\pi r l$ is the curved surface where $l$ is the slant length
rectangular prism (cuboid)	$lwh$	$2(lw + wh + lh)$	beware of open-top boxes