Surface area refers to the total area that the surface of a three-dimensional object occupies. It is a measure of how much-exposed area the object has. It’s used in various fields like engineering, architecture, and design to determine the material needed for constructing or covering objects.
The method for calculating surface area varies depending on the shape of the object. Here, we will explore the different surface area formulas used to determine the total area of the outer surfaces of three-dimensional geometric shapes.
Learning surface area formulas helps in quickly solving the questions appearing on exams as well as practical applications like calculating material requirements, understanding scientific processes, and addressing engineering problems.
Types of Surface Area
Surface Area is only calculated for 3-D figures. We can have two types of Surface Areas:
Lateral/Curved Surface Area
- It refers to the area of all the sides of a three-dimensional object, excluding the base(s).
- It specifically focuses on the "sides" or "walls" of the object.
Total Surface Area
- It refers to the sum of the areas of all the outer surfaces (including both the lateral surface and the bases).
- It covers the entire area of an object.
The table below highlights the key differences, formulas, and applications of Lateral Surface Area (LSA) and Total Surface Area (TSA):
| Aspect | Lateral Surface Area (LSA) / Curved Surface Area (CSA) | Total Surface Area (TSA) |
|---|---|---|
| Definition | The area of the curved or side surfaces of a figure. | The area of all surfaces of the figure, including the top, base, and sides. |
| Also Known As | Curved Surface Area | TSA |
| Formula (General Concept) | LSA = Area of Side Faces | TSA = LSA + Area of Top Surface + Area of Base Surface |
| Application | Used for objects with curved sides like cylinders, cones, etc. | Used for all 3D figures to determine the complete outer area. |
Formulas for Surface Area
Surface area formulas are given for the total surface area (TSA) and the lateral surface area (LSA). The total surface area includes the area of all surfaces of the object (base + sides), while the lateral surface area includes only the area of the sides, excluding the base.
There are various surface area formulas, and some of the key surface areas for important geometric figures are listed in the table below:
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Surface Area Formulas List
The following table contains the surface area formulas of different shapes
Shape | Figure | Lateral Surface Area (LSA) | Total Surface Area (TSA) |
|---|---|---|---|
| 4a2 | 6a2 | |
.png) | 2h(l+b) | 2(lb + lh + bh) | |
| 2πrh | 2πr(r + h) | |
| πrl | πr(l + r) | |
| 4πr2 | 4πr2 | |
| 2πr2 | 3πr2 | |
| 1/2 × (Base Perimeter) × (Slant Height) | LSA + Area of Base | |
| (Base Perimeter) × (Height) | LSA + 2(Area of Base) |
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Surface Area of Different Shapes in Detail
Below are the formulas, along with the definitions and figures, for the Lateral Surface Area (LSA) and Total Surface Area (TSA) of various 3D geometrical figures.
Surface Area Formula of Cube
A cube is a six-faced 3D shape in which all the faces are equal. A cube is a three-dimensional shape with several key characteristics:
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Formulas for the Surface Area of Cube a are given by:
- Lateral Surface Area (LSA) of Cube = 4a2
- Total Surface Area (TSA) of Cube = 6a2
Where a is the Side of a Cube.
To learn more about the formulas related to a Cube, here are the key ones:
Surface Area Formula of Cuboid
A Cuboid is a 3D figure in which opposite faces are equal. A cuboid, also known as a rectangular prism, is a 3D geometric shape very similar to a cube, but with some key differences:
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Formulas for the Surface Area are given by:
- Lateral Surface Area (LSA) of Cuboid = 2 × (hl + bh)
- Total Surface Area (TSA) of Cuboid = 2 × (hl + bh + bh)
Where:
- l is the length of the Cuboid
- b is the Breadth of Cuboid
- h is the Height of the Cuboid
To learn more about the formulas related to a Cuboid, here are the key ones:
Surface Area Formula of a Sphere
The sphere is a 3D figure that is similar to a real-life ball. A sphere is a three-dimensional, perfectly round object with several key characteristics:
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The formula for the Surface Area of a Sphere is:
Surface Area of Sphere = 4πr2
Where r is the Radius of the Sphere.
To learn more about the formulas related to a Sphere, here are the key ones:
Surface Area Formula of a Hemisphere
The hemisphere is a 3D figure that is half of the Sphere. It is created by slicing it through its center with a flat plane.
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The formula for the Area of a Hemisphere formula is:
- Curved Surface Area (CSA) of Hemisphere = 2πr2
- Total Surface Area (TSA) of Hemisphere = 3πr2
Where r is the Radius of the Sphere.
To learn more about the formulas related to a Hemi-sphere, here are the key ones:
Surface Area Formula of a Cylinder
A cylinder is a 3D figure with two circular bases and a curved surface.
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The formula for the Area of a Cylinder is:
- Curved Surface Area (CSA) of Cylinder = 2πrh
- Total Surface Area (TSA) of Cylinder = 2πr2 + 2πrh = 2πr(r+h)
Where:
- r is the Radius of the base of the Cylinder
- H is the Height of Cylinder
To learn more about the formulas related to a Cylinder, here are the key ones:
Surface Area Formula of a Cone
A cone is a 3D geometric shape with a circular base and a pointed edge at the top called the apex. A cone has one face and a vertex.
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The Formula for the Surface Area of the Cone is:
Curved Surface Area (CSA) of Cone = πrl
Total Surface Area (TSA) of Cone = πr(r + l)
Where:
- r is the Radius of the Base of the Cone
- l is the Slant Height of the Cone
To learn more about the formulas related to a Cone, here are the key ones:
Surface Area Formula of Pyramid
A pyramid is a 3D figure having triangular faces and a triangular base. It is a three-dimensional polyhedron with a polygonal base and triangular sides that meet at a common point called the apex.
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The formula for the Surface Area of a Pyramid is:
- Lateral Surface Area (LSA) of Pyramid = 1/2 × (Perimeter of Base) × Height
- Total Surface Area (TSA) of Pyramid = [1/2 × (Perimeter of Base) × Height] + Area of Base
To learn more about the formulas related to a Pyramid, here are the key ones:
Solved Examples of Surface Area Formulas
Example 1: Find the lateral surface of a Sphere with a radius of 4 cm.
Solution:
Given,
Radius of Sphere (r) = 4 cmFormula of Lateral Surface Area of Sphere = 4πr2
LSA = 4 × 3.14 × r × r = 4 × 3.14 × 4 × 4
LSA = 200.96 cm2
Example 2: Find the lateral surface of a Hemi-Sphere with a radius of 6 cm.
Solution:
Given,
Radius of Hemisphere (r) = 6 cmFormula of Lateral Surface Area of Hemi-Sphere = 2πr2
LSA = 2 × 3.14× r × r = 2 × 3.14 × 6 × 6
LSA = 226.08 cm2
Example 3: Find the Total surface of a Cube with a side of 10 m.
Solution:
Given,
Side of Cube (a) = 10 cmFormula of Total Surface Area of Cube = 6a2
TSA = 6 × a × a = 6 × 10 × 10
TSA = 600 m2
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