Prisms and pyramids are two types of 3D shapes often studied in geometry. Although they may look similar, they have unique characteristics that set them apart.
What is a Prism?
A prism is a three-dimensional shape that has two identical and parallel faces, called bases, and all other faces are rectangles. The bases can be any polygon (triangle, square, rectangle, pentagon, etc.). Prisms are named based on the shape of their bases.
Types of Prisms
- Triangular Prism
- Has triangular bases.
- Example: The shape of a tent or a Toblerone chocolate bar.
- Rectangular Prism (Cuboid)
- Has rectangular bases.
- Example: A typical shoebox or brick.
- Pentagonal Prism
- Has pentagonal bases.
- Example: The cross-section of certain crystals and architectural elements.
- Hexagonal Prism
- Has hexagonal bases.
- Example: The shape of a honeycomb cell in a beehive.
Properties of Prisms
- Faces: Prisms have two identical bases and several rectangular side faces.
- Edges: Prisms have more edges than pyramids. The number of edges is calculated by multiplying the number of sides of the base polygon by 3.
- Vertices: A prism has twice as many vertices as the number of sides on its base.
Formula for Volume of a Prism
The volume of a prism is determined by the area of its base and the height of the prism (the distance between the two bases).
Volume of a Prism = Base Area × Height
where:
- Base Area is the area of the prism’s base (e.g., for a triangular prism, it’s the area of the triangle).
- Height is the perpendicular distance between the two bases.
Example: For a triangular prism with a base area of 20 square centimeters and a height of 10 centimeters, the volume would be:
Volume = 20 cm² × 10 cm = 200 cm³
What is a Pyramid?
A pyramid is a 3D shape with a single polygonal base and triangular faces that meet at a common point called the apex. Unlike prisms, which have two bases, pyramids have only one, and all side faces are triangular.
Types of Pyramids
- Triangular Pyramid (Tetrahedron)
- Has a triangular base.
- Example: The Egyptian pyramids or certain dice used in board games.
- Square Pyramid
- Has a square base.
- Example: The Great Pyramid of Giza in Egypt.
- Pentagonal Pyramid
- Has a pentagonal base.
- Example: Some decorative structures and complex architectural designs.
Properties of Pyramids
- Faces: Pyramids have one polygonal base, and the number of triangular faces equals the number of sides on the base.
- Edges: The number of edges in a pyramid is calculated by adding the number of sides of the base to the total number of triangular faces.
- Vertices: Pyramids have one additional vertex at the apex, compared to the number of vertices on the base.
Formula for the Volume of a Pyramid
The volume of a pyramid is calculated using the area of its base and the height, but the formula is different from that of a prism. The height of the pyramid is the perpendicular distance from the apex to the center of the base.
Volume of a Pyramid = $\frac{1}{3}$ x Base Area x Height
where:
- Base Area is the area of the base polygon.
- Height is the perpendicular distance from the apex to the base.
Example: For a square pyramid with a base area of 16 square meters and a height of 9 meters, the volume would be:
Volume = $\frac{1}{3}$ x 16 m² x 9 m = 48 m³
Comparison: Prisms Vs Pyramids
The following table lists the key differences between a prism and a pyramid.
Fun Activity
For a hands-on learning experience, try building your own models of prisms and pyramids using paper, cardboard, or clay. Start with a base shape (triangle, square, pentagon, etc.) and build the faces from there. Count the edges, faces, and vertices, and then calculate the volume using the formulas given above.
Why Learn About Prisms and Pyramids?
Studying prisms and pyramids helps us understand the properties of 3D shapes, spatial relationships, and geometry. These shapes are commonly found in architecture, engineering, art, and even in nature. Recognizing the differences between them and knowing how to calculate their volumes are crucial skills in various fields such as construction, design, and mathematics.
FAQs on Prism and Pyramids
- In a right prism, the sides are perpendicular to the base, meaning the lateral faces are rectangles. In an oblique prism, the sides are slanted, and the lateral faces are parallelograms.
- The number of faces, edges, and vertices in a prism depends on the number of sides in its base:
- A prism with an n-sided polygon base has:
- Faces: n+2 (two bases and n lateral faces).
- Edges: 3n
- Vertices: 2n. For example, a triangular prism has 5 faces, 9 edges, and 6 vertices.
- A prism with an n-sided polygon base has:
- Triangular Pyramid: The base is a triangle.
- Square Pyramid: The base is a square.
- Rectangular Pyramid: The base is a rectangle.
- Pentagonal Pyramid: The base is a pentagon, and so on for other polygons.
- Regular Pyramid: The base is a regular polygon, and all triangular faces are congruent.
- A pyramid with an n-sided polygon base has:
- Faces: n+1 (one base and n triangular faces).
- Edges: 2n.
- Vertices: n+1 (the base vertices plus the apex). For example, a square pyramid has 5 faces, 8 edges, and 5 vertices.
- When you slice a prism parallel to its base, the cross-section is identical to the base in both shape and size.
- When you slice a pyramid parallel to its base, the cross-section is a smaller version of the base. As you move upward toward the apex, the cross-sections get smaller.