Pyramid

A pyramid is a 3D shape with a polygonal base and triangular faces that meet at a single point, the apex.

• The pyramid's height is the perpendicular distance from the base to the apex.
• Pyramids can have different base shapes (e.g., square, triangular) and are studied in geometry for their volume and surface area.

Pyramid Example

Here are some of the examples of Pyramid:

Types of Pyramid

Pyramids have different types based on their base shape, like squares, triangles, and pentagons. Each type has its special geometric features.

• Triangular Pyramid: A pyramid with a three-sided base, forming a point at the top, similar to a tent with a triangular base.
• Square Pyramid: A pyramid with a square bottom and sides that make a triangle shape, like the Great Pyramid of Giza.
• Rectangular Pyramid: A Pyramid in which the base is rectangular in shape and the side faces are triangular
• Pentagonal Pyramid: A pyramid with a base that has five sides, converging to a point at the top, seen in some ancient Mayan temples.
• Hexagonal Pyramid: A pyramid with a hexagon base and has 6 triangular faces.

Right Pyramid vs Oblique Pyramid

In a right pyramid, all triangular sides meet the base at right angles, while in an oblique pyramid, at least one side doesn't.

Regular vs Irregular Pyramid

A regular pyramid has equal sides and angles with a regular base shape, while an irregular pyramid can have varying sides and angles with an irregular base.

Pyramid Formulas

A pyramid is a 3D shape with a flat, polygonal base and triangular sides meeting at a point. Pyramid Formulas deal with the following two formulas:

• Volume of Pyramid
• Surface Area of Pyramid

Volume of a Pyramid

To find the volume of a pyramid, you take the area of its base, multiply it by the height, and then divide the result by 3.

V= 1/3 × Base Area × Height

Surface Area of a Pyramid

The formula for finding the surface area (A) of a pyramid is to add the area of its base to the sum of the areas of its triangular sides.

Area of Base (B) of Pyramid

• For a square base, use the formula: B = (side)²
• For a rectangular base, use: B = length × width
• For a triangular base, use: B = (1 / 2) × base length × height

The sum of Areas of Triangular Sides (T)

Add up the areas of each triangular side using:

T = ∑ [(1 / 2) × (base length of each side) × (slant height of each side)]

Finally, calculate the total surface area A by adding B and T.

Total surface area of Pyramid A = B + T

Net of a Pyramid

The net of a pyramid is a two-dimensional representation that, when folded, constructs the three-dimensional pyramid. It serves as a flattened layout showcasing the various surfaces of the pyramid, including the base and triangular faces. The edges on the net correspond to the connecting points of the pyramid's surfaces. This process of unfolding and folding helps visualize the spatial arrangement of the pyramid in a simpler form. Exploring nets is a valuable tool for comprehending the geometric structure of three-dimensional shapes.

Properties of Pyramid

The properties of the Pyramid are mentioned below:

• Base Shape: The base is a flat shape with straight sides, like a simple drawing.
• Triangular Sides: The sides are like triangles, connecting the corners of the base to a point at the top.
• Apex: The apex is just the very top point where all the triangles meet.
• Height: The height is how tall the pyramid is, measured straight down from the top to the bottom.
• Surface Area: The surface area is how much space the pyramid covers, including the bottom and the triangular sides.
• Volume: The volume is how much stuff can fit inside the pyramid, found by multiplying the base size by the height and then dividing by 3.
• Symmetry: The pyramid looks the same if you turn it around, whether you look from the top or the bottom.
• Diagonals: Lines connecting non-adjacent corners on the base.
• Euler's Formula: For a pyramid with a flat shape at the bottom, the number of corners (V), sides (E), and faces (F) satisfy the rule: V - E + F = 2.

Also, Check

• Volume of a Pyramid Formula
• Surface Area of a Pyramid Formula
• Hexagonal Pyramid Formula 

Solved Examples on Pyramid

Example 1: Find the volume of a triangular pyramid if the base area is 36 cm² and the height is 12 cm.
Solution:

Given:

Base area of the triangular pyramid = 36 cm²
Height = 12 cm
The volume of the triangular pyramid is given by the formula: Volume = (1/3) × (Base area) × (Height).

Substitute the values into the formula:
Volume = (1/3) × 36 × 12
Volume = (1/3) × 432
Volume = 144 cm³

Therefore, the volume of the triangular pyramid is 144 cm³.

Example 2: Determine the total surface area of a pentagonal pyramid if the slant height is 8 cm, and the apothem (distance from the center to the midpoint of a side) is 6 cm.
Solution:

Given:

Slant height = 8 cm
Apothem = 6 cm

The total surface area of the pentagonal pyramid is given by the formula: TSA = (1/2) × Perimeter of the base × Slant height + Area of the base.
The perimeter of the base, P = 5 × side length.

P = 5 × 8
P = 40 cm

The area of the base, B = (1/2) × P × Apothem.

B = (1/2) × 40 × 6
B = 120 cm²

Now, substitute the values into the total surface area formula:
TSA = (1/2) × 40 × 8 + 120
TSA = 160 + 120
TSA = 280 cm²

Hence, the total surface area of the pentagonal pyramid is 280 cm².