Polyhedron

Polyhedron ☰

A polyhedron is a three-dimensional geometric solid object that is formed by connecting flat polygonal faces together along their edges. Each face is a polygon, which is a two-dimensional shape with straight sides. The corners where the faces meet are called vertices, and the straight lines that connect these vertices are called edges.


Polyhedra can be classified based on various properties:

⠐ Regular polyhedra (Platonic solids): There are five Platonic solids, in which all the faces are congruent, regular polygons, and the same number of faces meet at each vertex. They are:

Tetrahedron: 4 faces, each a triangle (3 equilateral triangles meeting at each vertex)

Cube (Hexahedron): 6 faces, each a square (3 squares meeting at each vertex)

Octahedron: 8 faces, each a triangle (4 equilateral triangles meeting at each vertex)

Dodecahedron: 12 faces, each a pentagon (3 regular pentagons meeting at each vertex)

Icosahedron: 20 faces, each a triangle (5 equilateral triangles meeting at each vertex)

⠐ Semi-regular polyhedra (Archimedean solids): These are polyhedra with regular polygonal faces, but not all faces are the same. There are 13 Archimedean solids, and they are formed by truncating or expanding the Platonic solids.

Prisms: Prisms are polyhedra with two congruent, parallel polygonal faces (bases) and a set of rectangular faces connecting the corresponding edges of the bases.

Pyramids: Pyramids are polyhedra with a polygonal base and triangular faces that meet at a common vertex (apex).

Other polyhedra: There are many other types of polyhedra that don't fall into the categories above, including irregular polyhedra with non-uniform face shapes or vertex configurations.

The properties of a polyhedron can be described using the Euler's formula, which relates the number of faces (F), vertices (V), and edges (E) in a polyhedron:
\( V-E+F=2 \)

This formula holds for any convex polyhedron, regardless of its specific shape or arrangement of faces.

Prisms ☰

A prism is a special type of polyhedron, a three-dimensional geometric solid with flat faces. Prisms are characterized by two congruent, parallel polygonal faces called bases, which are connected by a set of rectangular faces. The shape of the bases determines the type of the prism.

Here are some key properties and characteristics of prisms:

Bases: The bases are congruent polygons, meaning they have the same shape and size. They are parallel to each other and determine the overall shape of the prism. Common prisms include triangular, rectangular, and pentagonal prisms, based on the shapes of their bases.

Lateral faces: The lateral faces are the rectangular faces that connect the corresponding edges of the bases. The number of lateral faces is equal to the number of sides in the base polygon. Lateral faces are always parallel to each other.

Edges: A prism has two types of edges: base edges and lateral edges. Base edges are the edges of the base polygons, while lateral edges connect the vertices of the bases. The number of edges in a prism is twice the number of sides in the base polygon plus the number of lateral edges.

Vertices: A prism has two sets of vertices—one set for each base. The number of vertices in a prism is twice the number of vertices in the base polygon.

Right and oblique prisms: A right prism is a prism where the lateral edges are perpendicular to the base polygons, meaning the lateral faces are also perpendicular to the base polygons. In an oblique prism, the lateral edges are not perpendicular to the base polygons, causing the lateral faces to slant.

Volume: The volume of a prism is calculated by multiplying the area of the base \((A)\) by the height \((h)\) of the prism (the perpendicular distance between the bases). The formula for the volume is: \( \text{Volume}=A \cdot h \)

Surface area: The surface area of a prism is the sum of the areas of its faces, which includes the areas of the bases and the lateral faces. To find the surface area, calculate the area of the base polygon, the area of one lateral face, and then multiply and sum the areas accordingly.

A prism whose base is a parallelogram is called a parallelepiped. Opposite faces of a parallelepiped parallel and congruent. A straight parallelepiped with a rectangular seat is called a rectangular parallelepiped. Total surface area of a rectangular parallelepiped with length a, width b, height c
It is calculated by the formula \( S = 2 ( ab + ac +bc ) \).

Surface area of the prism ☰

The surface area of a prism is the total area of all its faces, including the bases and lateral faces. To find the surface area, you'll need to calculate the area of each face and then sum those areas. Here's a step-by-step approach to find the surface area of a prism:

Determine the base polygon: Identify the shape of the base polygon (e.g., triangle, rectangle, pentagon). This shape will be used to calculate the area of the bases.

Calculate the area of the base polygon: Use the appropriate formula for the area of the base polygon. For instance:
⠐ Triangle: \( \frac{1}{2} \cdot \text{ base } \cdot \text{ height } \)
⠐ Rectangle: \( \text{ length } \cdot \text{ width } \)
⠐ Regular polygon (n sides, side length s): \( \frac{n \cdot S^2 }{4 \cdot tan ( \frac{ \pi }{n} )} \)

Calculate the area of both bases: Since the bases are congruent, multiply the area of one base by 2 to find the total area of both bases.

Calculate the lateral surface area: To do this, you'll need the slant height \((l)\) of the lateral face and the base perimeter \((P)\). The slant height is the height of the rectangle forming the lateral face. The base perimeter is the sum of the lengths of all the base edges. Lateral face area formula given by:
\( \text{Lateral face area}=P \cdot l \).
Note: For a right prism, the slant height is the same as the height (h) of the prism, which is the perpendicular distance between the bases.

Sum the areas: $$ \text{Surface area} = \text{Total area of both bases} + \text{Total area of lateral faces} $$

The volume of a prism ☰

The volume of a prism is the amount of space it occupies in three dimensions. To find the volume of a prism, you need to determine the area of its base and multiply it by the height of the prism. The height of the prism is the perpendicular distance between the two bases. Here's a step-by-step approach to find the volume of a prism:

Determine the base polygon: Identify the shape of the base polygon (e.g., triangle, rectangle, pentagon). This shape will be used to calculate the area of the base.

Calculate the area of the base polygon: Use the appropriate formula for the area of the base polygon. For instance:
⠐ Triangle: \( \frac{1}{2} \cdot \text{ base } \cdot \text{ height } \)
⠐ Rectangle: \( \text{ length } \cdot \text{ width } \)
⠐ Regular polygon (n sides, side length s): \( \frac{n \cdot S^2 }{4 \cdot tan ( \frac{ \pi }{n} )} \)

Determine the height of the prism: The height \((h)\) of the prism is the perpendicular distance between the two bases. For a right prism, the height is the same as the length of the lateral edges.

Calculate the volume: Multiply the area of the base \((A)\) by the height \((h)\) of the prism: \( \text{Volume}=A \cdot h \)

Let's look at an example using a right triangular prism:
Suppose we have a right triangular prism with a base triangle having sides of length 3, 4, and 5 units and a height of 6 units.

• The base is a triangle.
• The base triangle is a right-angled triangle with legs of length 3 and 4. Its area is \( \frac{1}{2} \cdot 3 \cdot 4 = 6 \) square units.
• The height of the prism is 6 units (given).
• The volume of the prism is \( 6 \text{(base area)} \cdot 6 \text{(height)}=36 \) cubic units.
  The volume of this right triangular prism is 36 cubic units.

Pyramid. Lateral surface area and total surface area of a pyramid. ☰

A pyramid is a three-dimensional polyhedron with a polygonal base and triangular faces that meet at a single point called the apex. The most famous example of a pyramid is the Egyptian pyramids, which have a square base and four triangular faces. However, pyramids can have different bases, such as triangular, rectangular, or pentagonal, among others. The triangular faces are called lateral faces, and their edges are called lateral edges.

Lateral Surface Area of a Pyramid:
The lateral surface area of a pyramid is the sum of the areas of its lateral faces. To find the lateral surface area, you can use the following formula:
\( \text{Lateral Surface Area} = \frac{1}{2} \cdot P \cdot L \), where \(P\) is the perimeter of the base, and \(L\) is the slant height of the pyramid. The slant height is the height of a triangular face when measured along its slope from the apex to the midpoint of an edge of the base.

Total Surface Area of a Pyramid:
The total surface area of a pyramid includes the area of its base and the lateral surface area. To find the total surface area, use this formula:
\( \text{Total Surface Area} = \text{Base Area} + \) \( \text{Lateral Surface Area} \).

So, depending on the shape of the base, you'll need to calculate the base area accordingly. For example, if you have a square pyramid, the base area can be calculated using the formula:
\( \text{Base Area}=a^2 \), where \(a\) is the length of a side of the square base.

For a triangular pyramid, also known as a tetrahedron, the base area can be calculated using Heron's formula or half base times height, depending on the information you have.

In summary, to find the total surface area of a pyramid, calculate the base area and lateral surface area separately, then add them together.

Volume of a pyramid ☰

The volume of a pyramid is the amount of space it occupies in three dimensions. To calculate the volume of a pyramid, you need to know the area of its base and its height. The height of a pyramid is the perpendicular distance between the apex and the base, which is different from the slant height.

Here's the formula to calculate the volume of a pyramid:
\( \text{Volume}=\frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} \)

Let's break down this formula and discuss it in more detail:

Base Area (B): This is the area of the polygonal base of the pyramid. Depending on the shape of the base, you'll need to calculate the base area accordingly. For example, if you have a square pyramid, you can calculate the base area using the formula: \(a^2\), where '\(a\)' is the length of a side of the square base. For a triangular pyramid (tetrahedron), you can calculate the base area using Heron's formula or half base times height, depending on the information you have.

Height (h): The height of the pyramid is the perpendicular distance between the apex and the base. It's important not to confuse this with the slant height, which is the height of a triangular face when measured along its slope from the apex to the midpoint of an edge of the base.

\( \frac{1}{3} \): This fraction represents the relationship between the volume of a pyramid and the volume of a prism that has the same base area and height. In other words, the volume of a pyramid is one-third the volume of a prism with the same base area and height.

Once you have calculated the base area and determined the height, simply plug the values into the volume formula and solve for the volume.

Example:
Let's say you have a square pyramid with a base side length of 4 units and a height of 6 units. To calculate the volume:

Calculate the base area (B): \(B=a^2=4^2=16 \) square units

Use the formula: $$ \text{Volume}=\frac{1}{3} \cdot \text{Base Area} \cdot \text{Height} =\frac{1}{3} \cdot 16 \cdot 6=32 \text{ cubic units} $$

So, the volume of the square pyramid is 32 cubic units.