Geometry and Trigonometry - Area and volume of 3D shapes

Prisms and Uniform Cross-sections: A prism is a 3D solid with a constant cross-section throughout its length. Imagine a 2D shape, like a triangle or hexagon, being stretched along a straight line. To find its volume, you simply multiply the area of this base shape by the height or length of the prism.

Cylinders as Circular Prisms: A cylinder consists of two parallel circular bases connected by a curved surface. If you were to 'unroll' the curved side of a cylinder, it would form a flat rectangle. The length of this rectangle corresponds to the circumference of the circle ($2\pi r$), and its width is the height ($h$) of the cylinder.

Pyramids and Cones: Unlike prisms, pyramids and cones taper to a single point called the apex. Visually, a pyramid has a polygon base (like a square or triangle) and flat triangular sides, while a cone has a circular base and a smooth curved side. Their volume is always $\frac{1}{3}$ of a prism or cylinder with the same base and height.

The Relationship in Cones (Slant Height): In a right cone, the vertical height ($h$), the radius of the base ($r$), and the slant height ($l$) form a right-angled triangle. This allows us to use the Pythagorean theorem, $l = \sqrt{r^2 + h^2}$, to find missing dimensions needed for surface area or volume calculations.

Spheres and Hemispheres: A sphere is a perfectly round 3D object where every point on the surface is the same distance ($r$) from the center. A hemisphere is exactly half of a sphere. When calculating the surface area of a solid hemisphere, you must remember to add the area of the circular flat face ($\pi r^2$) to the curved half-surface ($2\pi r^2$).

Surface Area vs. Volume: Volume measures the capacity or the amount of 3D space an object occupies, measured in cubic units like $cm^3$. Surface area is the total area of all the outer faces of the object, measured in square units like $cm^2$. For composite shapes, volume is additive, but surface area requires careful subtraction of the faces that are touching or hidden.

Units and Conversions: Always ensure dimensions are in the same units before calculating. Remember that for area, $1\text{ m}^2 = 10,000\text{ cm}^2$ ($100 \times 100$), and for volume, $1\text{ m}^3 = 1,000,000\text{ cm}^3$ ($100 \times 100 \times 100$).