Pyramid

Pyramid is a three-dimensional geometric shape that features a flat base and triangular sides converging at a single apex. Studying its properties involves understanding base shapes, symmetry, and formulas for volume and surface area. In this article, the complexities of pyramids are mentioned with the solved examples and frequently asked questions in the end.

What is Pyramid?

A pyramid is a 3D shape with a flat shape at the bottom and triangular sides that meet at a point on top, called the apex. The height is how tall the pyramid is from the bottom to the top. To find its surface area, add the base and triangular side areas. Volume is calculated by multiplying the base area by the height and dividing by 3. People in geometry often study pyramids, using formulas to figure out their size, and they’re different from prisms.

Pyramid Definition

A pyramid is a three-dimensional geometric shape characterized by a flat, polygonal base and triangular sides that converge at a single point called the apex. The height is the vertical distance from the base to the apex.

Figure of Pyramid

Pyramid Example

Here are some of the examples of Pyramid:

• Pyramids of Egypt: Iconic structures like the Great Pyramid of Giza represent ancient Egyptian architecture.
• Pyramid Pastry: Creative pastries shaped like pyramids showcase the geometric form in culinary arts.
• Toys: Some Rubik cubes and rock-a-stack toys feature pyramid shapes, adding variety to toy designs.
• Tent: Camping tents often adopt a pyramid-like structure with wooden logs forming triangular faces.
• Tower: Roadside telephone towers commonly exhibit a pyramid shape with converging metallic bars.
• Temples: The shikhara or Shikhar (top) of many temples showcases a pyramid shape with triangular faces and a square base.
• Watermelon: Sliced watermelons often take on pyramid-shaped wedges for convenient serving.
• Wet Floor Sign: Common in public spaces, wet floor signs often have a three-dimensional pyramid-like shape.
• Chocolate: Some chocolate bars, like Toblerone, feature pyramid-shaped chocolate blocks attached to each other.

Properties of Pyramid

The properties of 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.

Types of Pyramid

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

Rectangular Pyramid

Rectangular Pyramid is the one whose base is rectangular in shape and side faces are triangular in nature

Learn, Rectangular Pyramid

Square Pyramid

A pyramid with a square bottom and sides that make a triangle shape, like the Great Pyramid of Giza.

Learn, Square Pyramid

Triangular Pyramid

A pyramid with a three-sided base, forming a point at the top, similar to a tent with a triangular base.

Learn, Triangular Pyramid

Pentagonal Pyramid

A pyramid with a base that has five sides, converging to a point at the top, seen in some ancient Mayan temples.

Learn, Pentagonal Pyramid

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 deals with following two formulas

• Volume of Pyramid Formula
• Surface Area of Pyramid Formula

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

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.

Net of Rectangular Pyramid

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.

Pyramid – Practice Questions

Here, are some following practice questions to solve.

Q1. A triangular pyramid has a base with sides of length 9 cm, 12 cm, and 15 cm. If the height from the apex to the center of the base is 8 cm, find the volume of the pyramid.

Q2. A square pyramid has a slant height of 10 cm, and each side of the base measures 6 cm. Calculate the total surface area of the pyramid.

Q3. A hexagonal pyramid has a regular hexagonal base with a side length of 7 cm. If the apothem (distance from the center to the midpoint of a side) is 6 cm, find the volume of the pyramid.

Q4. Find the lateral surface area of a pentagonal pyramid with a slant height of 12 cm and a regular pentagonal base with sides of length 5 cm.

Q5. A cylindrical pyramid has a circular base with a radius of 5 cm, and its height is 14 cm. Calculate the volume of the pyramid.

Q6. A triangular pyramid has a base with sides of length 10 cm, 24 cm, and 26 cm. If the altitude from the apex to the base is 9 cm, find the volume of the pyramid.

Pyramid – FAQs

1. What is a Pyramid?

A pyramid is a 3D shape that has a flat bottom (called the base) and sides that meet at a point on top (called the apex).

2. How do you figure out the Space inside a Pyramid?

To find the space inside a pyramid, you use the height and the area of the base. You multiply the base area by the height and then divide by 3.

3. What is the Surface Area of a Pyramid?

The surface area of a pyramid is the sum of the base area and the areas of its triangular sides.

4. What is a Net in Math, and how does it relate to a Pyramid?

A net is sort of a flat pattern of a 3D shape. For a pyramid, it’s a way of showing how the surfaces would look if you unfolded and flattened it.

5. How many Edges does a Square Pyramid have?

A square pyramid has eight edges. Four are on the base, and four connect the top point to each corner of the base.

6. How is a Pyramid different from a Prism?

A pyramid has a pointy top and triangular sides, while a prism has flat sides and the same shape at both ends.

7. Can a Pyramid have a Circle at the Bottom instead of a Square or Triangle?

Yes, a pyramid can have a circular base.

8. How do you find the Slant Height of a Pyramid?

To find the slant height, you use the base length, height, and a bit of math called the Pythagorean Theorem. It helps figure out the slant height.