Three-Dimensional Coordinate System
The three-dimensional coordinate system extends the 2D plane into 3D space by adding a third axis, typically labeled z. Points in 3D space are represented by ordered triples $(x, y, z)$.
Distance Between Two Points in 3D Space
The distance formula in 3D is an extension of the 2D Pythagorean theorem:
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$
Where $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are the coordinates of the two points.
Calculate the distance between points A(1, 2, 3) and B(4, 6, 8):
$d = \sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2}$ $= \sqrt{3^2 + 4^2 + 5^2}$ $= \sqrt{9 + 16 + 25}$ $= \sqrt{50} = 5\sqrt{2}$
Midpoint in 3D Space
The midpoint formula in 3D is a straightforward extension of the 2D version:
$M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$
Find the midpoint of the line segment connecting A(1, 2, 3) and B(4, 6, 8):
$M = (\frac{1+4}{2}, \frac{2+6}{2}, \frac{3+8}{2}) = (2.5, 4, 5.5)$
Volume and Surface Area of 3D Solids
Right Pyramid
A right pyramid has a polygonal base and triangular faces that meet at a point (apex) directly above the center of the base.
Volume: $V = \frac{1}{3} \times B \times h$, where B is the area of the base and h is the height.
For a square pyramid with base side 6 units and height 8 units: $V = \frac{1}{3} \times 6^2 \times 8 = 96$ cubic units
Right Cone
A right cone has a circular base with a point (apex) directly above its center.
Volume: $V = \frac{1}{3}\pi r^2 h$, where r is the radius of the base and h is the height.
Surface Area: $SA = \pi r^2 + \pi r l$, where l is the slant height.
For a cone with radius 3 units and height 4 units: $V = \frac{1}{3}\pi \times 3^2 \times 4 = 12\pi$ cubic units
Remember that the slant height of a cone forms the hypotenuse of a right-angled triangle with the radius and height as the other two sides. Thus, the slant height of a cone is given by $l^2 = r^2 + h^2$.
Sphere
Volume: $V = \frac{4}{3}\pi r^3$, where r is the radius.
Surface Area: $SA = 4\pi r^2$
Hemisphere
Volume: $V = \frac{2}{3}\pi r^3$
Surface Area: $SA = 3\pi r^2$ (including the circular base)
Remember that a hemisphere is half of a sphere, so its volume is half that of a sphere, but its surface area includes the circular base.
