๐ What is a Pyramid?
A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle, called a lateral face. The surface area of a pyramid is the total area of all its faces, including the base.
- ๐ Definition: A solid figure with a polygonal base and triangular faces that meet at a common point (apex).
- โจ Formula: Surface Area = Base Area + (1/2) * Perimeter of Base * Slant Height. Mathematically: $SA = B + \frac{1}{2}Pl$, where $B$ is the base area, $P$ is the perimeter of the base, and $l$ is the slant height.
- โฐ๏ธ Faces: Pyramids have one base and triangular lateral faces.
๐ What is a Prism?
A prism is a polyhedron with two congruent and parallel faces (bases) and whose other faces (lateral faces) are parallelograms. The surface area of a prism is the sum of the areas of all its faces.
- ๐ Definition: A solid figure with two parallel and congruent bases connected by rectangular or parallelogram lateral faces.
- โ Formula: Surface Area = 2 * Base Area + Perimeter of Base * Height. Mathematically: $SA = 2B + Ph$, where $B$ is the base area, $P$ is the perimeter of the base, and $h$ is the height of the prism.
- ๐งฑ Faces: Prisms have two bases and rectangular or parallelogram lateral faces.
๐ Pyramid vs. Prism: Surface Area Comparison
| Feature |
Pyramid |
Prism |
| Bases |
One base |
Two parallel and congruent bases |
| Lateral Faces |
Triangles |
Rectangles or Parallelograms |
| Apex |
Has an apex (a single point where lateral faces meet) |
No apex |
| Surface Area Formula |
$SA = B + \frac{1}{2}Pl$ (B = base area, P = base perimeter, l = slant height) |
$SA = 2B + Ph$ (B = base area, P = base perimeter, h = height) |
๐ก Key Takeaways
- ๐ Bases: Pyramids have one base, while prisms have two.
- ๐งฎ Lateral Faces: The lateral faces of pyramids are triangles, while those of prisms are rectangles or parallelograms.
- ๐งช Apex: Pyramids have an apex, while prisms do not. This fundamentally changes how you calculate the overall surface area.
- ๐ง Formulas: Remember to use the correct formula based on whether you are working with a pyramid or a prism. Pay close attention to slant height vs. height!