Pyramid shape formula for volume surface area and properties
In Geometry, a pyramid is a polyhedron formed by joining a polygonal base and a point, known as apex or vertex. Each base edge and apex forms a triangle known as the lateral faces. It is a three-dimensional geometric shape with a polygon base. A pyramid with an ‘x’ sided base has (x +1 ) vertices, (x + 1 ) faces, and (2x) edges. A pyramid is generally described by the shape of its base. For example, a triangular pyramid has a triangular base whereas a hexagonal pyramid has a hexagonal base.
Pyramid Image
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Parts of 3D Triangle Shape- Pyramid
Apex
An apex is the vertex of a pyramid at its top.
Edge
The line segments formed by joining two intersecting faces are the edges of a pyramid.
Lateral Face
The lateral faces of a pyramid shape are congruent isosceles triangles.
Base
A base is the bottom of the pyramid shape.
Real-Life Examples of Pyramid
Following are some of the real-life examples of pyramid
Pyramid Toy
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Pyramid Pastry
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Pyramid of Egypt
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Pyramid Chocolate
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Volume And Surface Area of Pyramid
The formula of volume of a pyramid shape is given as:
Volume = 1/3 \[\times \] Base Area \[\times \] Height |
Surface Area of Pyramid
The two parts of a surface area of the pyramid are:
Base Area ( Area of the base)
Lateral Area (Area of the side face)
When all the side faces of a pyramid shape are the same.
½ \[\times \] Base Area \[\times \] Perimetre \[\times \] Slant Height |
When all the side faces of a pyramid are different.
Base Area + Lateral Area |
Types of Pyramid
Following are the different types of the pyramid:
Square - Based Pyramid
A square-based pyramid is a three-dimensional object with a square base. It has four triangular faces/sides, 5 vertices, and 8 edges. The Egyptian pyramid is the best example of a square-based pyramid.
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A square-based pyramid is also known as a pentahedron because it has a total 5 faces- Penta means 5 in Latin. Out of 5 faces, 4 faces are triangular in shape and 1 face ( the polygon on which it rests upon) is square in shape.
If all the four triangles of a square-based pyramid are equilateral with equal sides and angles, then the square -based pyramid is referred to as Johnson Solid.
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Triangular Based Pyramid
A triangular-based pyramid is a solid object formed by joining a triangular base to a point known as apex or vertex. This creates 4 faces/sides, each of which is triangular in shape. It has 4 vertices and 6 edges.
A triangular based pyramid is also known as ‘Tetrahedron’ because the four faces of a triangular pyramid are triangular in shape.
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Pentagonal Pyramid
In Geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which five triangular faces are erected that meet at a point known as the apex or vertex.
The regular pentagonal pyramid is a pyramid with a base that is a regular pentagon. Also, this pyramid has 5 lateral faces that are equilateral triangles. A pentagonal pyramid is one of Johnson's solid.
It has a total of 6 faces, six vertices and 10 edges.
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Hexagonal Pyramid
In Geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which six isosceles triangles are erected that meet at a point known as the apex/vertex. A hexagonal pyramid has 7 faces, 7 vertices, and 12 edges.
Hexagonal pyramid is also known as heptahedron because it has 7 faces.
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Right V/S Oblique Pyramid
These types of pyramids tell where the top (apex) of the pyramid is. When the apex or vertex of a pyramid is found directly above the center of its base, then it is known as a regular pyramid, else it is known as an oblique pyramid.
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Regular V/S Irregular Pyramid
These types of pyramids tell about the base of the shape. When the base of a pyramid is a regular polygon, then it is considered a regular pyramid, else it is known as an irregular pyramid.
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Pyramid Questions With Examples
Find the volume of a regular square pyramid with base side 20 cm and altitude 15 cm?
Solution:
Given
Base Side = 20 cm
Accordingly, base area will be 20 \[\times \] 20 = 400 cm2
Altitude = 15 cm
The volume of a regular square pyramid can be calculated using the following formula:
⅓ \[\times \] (Base Area) \[\times \] Height
Using the formula, we get
⅓ \[\times \] 400 \[\times \] 15
= 2000
Hence, the volume of a given regular square pyramid is 2000 cm3.
What is the area of a regular square pyramid with base side 5 in and altitude 8 in?
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Solution:
Area of Base = 5 \[\times \] 5 = 25 cm2
Area of Lateral Surface = ½ \[\times \] 5 \[\times \] 8 = 20 cm2
Area of 4 sides faces = 20 + 20 + 20 + 20 = 80 cm2
Note: There are 4 identical lateral surface faces. Therefore we have counted lateral surface area 4 times.
The area of a regular square pyramid can be calculated using the formula
Surface Area = Area of Base + Area of Lateral Surface
By substituting the values in the above surface area of the pyramid formula, we get:
25 + 80
= 105 cm2
The surface area of a regular square pyramid is 105 cm2
Did You Know?
A pyramid is defined as a polyhedron that joins a polygon base (such as a triangle or square) to a point known as the apex or vertex using triangles.
The tetrahedron is the other name of a triangular pyramid. A tetrahedron has 4 triangular faces, with three faces meeting at each point known as the vertex.
An octahedron has 8 faces of an equilateral triangle, with four faces meeting at each point known as the vertex.
An icosahedron has 20 faces of identical equilateral triangles. An icosahedron pyramid has 12 vertices and 30 edges.
The great Pyramid that was built by Ancient Egyptians several years ago is an example of a square pyramid.
FAQs on Understanding the Pyramid Shape in Geometry
1. What is a pyramid shape in Maths?
A pyramid is a three-dimensional solid with a polygonal base and triangular faces that meet at a single point called the apex. In geometry, a pyramid has:
- A base (square, triangle, rectangle, etc.)
- Triangular lateral faces
- A common vertex called the apex
2. What is the formula for the volume of a pyramid?
The volume of a pyramid is given by V = (1/3) × Base Area × Height. Here:
- Base Area (B) is the area of the base shape.
- Height (h) is the perpendicular distance from the base to the apex.
3. How do you find the surface area of a pyramid?
The total surface area of a pyramid equals the base area plus the area of all triangular faces. The formula is Total Surface Area = Base Area + Lateral Surface Area.
- Base Area depends on the shape of the base.
- Lateral Surface Area = (1/2) × perimeter of base × slant height.
4. What is a regular pyramid?
A regular pyramid is a pyramid whose base is a regular polygon and whose apex is directly above the center of the base. This means:
- All base sides are equal.
- All triangular faces are congruent.
- The slant edges are equal in length.
5. What is the difference between a prism and a pyramid?
The main difference is that a prism has two parallel bases, while a pyramid has only one base and an apex. Key differences include:
- Prism: Two identical parallel bases.
- Pyramid: One base and triangular faces meeting at a point.
- Volume of prism = Base Area × Height.
- Volume of pyramid = (1/3) × Base Area × Height.
6. How many faces, edges, and vertices does a square pyramid have?
A square pyramid has 5 faces, 8 edges, and 5 vertices. Specifically:
- Faces: 1 square base + 4 triangular faces = 5
- Edges: 4 base edges + 4 lateral edges = 8
- Vertices: 4 base corners + 1 apex = 5
7. What is the slant height of a pyramid?
The slant height of a pyramid is the distance from the apex to the midpoint of a base edge along a triangular face. It is used to calculate lateral surface area. In a regular square pyramid, slant height (l), height (h), and half base side (a/2) form a right triangle, so:
- l² = h² + (a/2)²
8. How do you calculate the volume of a square pyramid?
The volume of a square pyramid is V = (1/3) × a² × h, where a is the base side and h is the height. Steps:
- Step 1: Find base area = a².
- Step 2: Multiply by height h.
- Step 3: Multiply by 1/3.
9. What are the types of pyramids in geometry?
Pyramids are classified based on the shape of their base. Common types include:
- Triangular pyramid (tetrahedron)
- Square pyramid
- Rectangular pyramid
- Pentagonal pyramid
- Hexagonal pyramid
10. Where are pyramid shapes used in real life?
Pyramid shapes are used in architecture, engineering, and design because of their structural stability. Real-life examples include:
- The Egyptian pyramids
- Glass pyramid at the Louvre Museum
- Roof structures and monuments
