Surface Area Formulas: Definitions, Solved Problems, Examples

Surface Area Formulas: The amount of the plane in two dimensions occupied by a surface is called the area of the surface. Each of the solid shapes – cube, cuboid, cylinder, sphere, cone, triangular prism, hemisphere etc., has one or two or three and, in some specific solids, more than three faces. The faces are either curved or plane. In this article, we shall discuss the formulas used to measure the surface areas of some solids.

The lateral surfaces and the base surface of any three-dimensional object are the same. The sum of the lateral/curved surface area and the base surface area is the total surface area. Continue reading this article to know more about Surface Area Formulas, Definitions, Solved Problems, Examples, etc.

Definition of Surface Area

The total area occupied by an object’s surfaces is referred to as its surface area. 

The area of the surface is divided into two categories:

1. Curved surface area (CSA) and Lateral surface area (LSA)
2. Total surface area (TSA)

Curved Surface Area

The curved surface area is defined as the area of only curved surfaces of an object, leaving the circular top and base.

We can see the curved surface areas in the case of solids like a sphere, cylinder, and cone. The solid shapes like cube, cuboid, triangular and square prism, etc., do not have a curved surface area. 

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Lateral Surface Area

The lateral surface of an object is all the sides of the object, excluding its base and top. The lateral surface area is the area of the lateral surface.

Total Surface Area

The total surface area is made up of the lateral surface area, as well as the base and top sections. 

In the case of cube, cuboid, triangular and square prism, the total surface area is the sum of all the areas of the flat surfaces. In the case of a sphere, the total surface area is the area of its only curved surface. In the case of a cylinder, the total surface area is the sum of its curved surface area and two flat surfaces at the top and the bottom. In the case of a cone, the total surface area is the sum of its curved surface area and the flat surface at the bottom. 

Surface Area of a Cube

In the three-dimensional plane, a cube is a three-dimensional box-like figure. The cube has six equal squared-shaped faces. At a \({\rm{9}}{{\rm{0}}^{\rm{o}}}\) angle, each face touches the other. At the same vertex, three sides of the cube intersect.

Lateral Surface Area of Cube

The sum of all the cube’s side faces is the cube’s lateral surface area. Because a cube has four side faces, its lateral area is equal to the sum of its areas.
The lateral surface area of a cube \(({\rm{LSA}}) = 4{a^2}\) Where \(“a”\) is the length of the side.

Total Surface Area of Cube

The overall surface area of the cube will be equal to the sum of the base area and the area of the cube’s vertical surfaces because there are six surfaces in all.
The total surface area \(({\rm{TSA}}) = 6{a^2}\) Where \(“a”\) is the length of the side.

Surface Area of a Cuboid

In the three-dimensional plane, a cuboid is a three-dimensional box-like figure.

Lateral Surface Area of Cuboid

The lateral surface area and the overall surface area of a cuboid can be expressed in terms of two different categories of area. 

The lateral surface area of a cuboid \(({\rm{LSA}}) = 2h(l + b)\,{\rm{sq}}{\rm{.units}}\)

Total Surface Area of Cuboid

The total surface area of the cuboid is calculated by summing the areas of all \(6\) faces. The lateral surface area is calculated by calculating the area of each face except the base and top. The total surface area and the lateral surface area are both important.
 The total surface area of a cuboid \(({\rm{TSA}}) = 2(lh + lb + bh)\,{\rm{sq}}{\rm{.units}}\)

Surface Area of a Cylinder

A cylinder is a three-dimensional solid object made up of two circular bases joined by a curved face. We can express the curved surface area as well as the overall surface area of a cylinder because it has a curved surface. The surface area of a cylinder is given as: if the radius of the base of the cylinder is \(r\) and the height of the cylinder is \(h,\) the surface area of the cylinder is:

Total surface area of a cylinder \(({\rm{TSA}}) = 2\pi r(h + r)\,{\rm{sq}}{\rm{.units}}\)

Curved surface area of a cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)

Surface Area of a Cone

The area occupied by the surface of a cone is known as its surface area. A cone is a three-dimensional shape with a circular base. This signifies that the base has a radius and a diameter. We can express the curved surface area as well as the overall surface area of a cone because it has a curved surface. If the radius of the cone’s base is \(r\) and the cone’s slant height is \(l.\)

Curved surface area of a cone \(({\rm{CSA}}) = \pi rl\,{\rm{sq}}{\rm{.units}}\)

Total surface area of a cone \(({\rm{TSA}}) = \pi r(r + l)\,{\rm{sq}}{\rm{.units}}\)

Surface Area of a Sphere

A sphere, like a circle, is a three-dimensional solid object with a circular structure. The surface area of a sphere refers to the area covered by the sphere’s outer surface. The entire area of the faces that surround a sphere is its surface area. A sphere’s surface area is measured in square units.

The total surface area of a sphere \(=\) curved surface area of a sphere.
The surface area of a sphere\( = 4\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)

Surface Area of a Hemisphere

A hemisphere is a three-dimensional form created by cutting a sphere along a plane that passes through its centre. A hemisphere, in other terms, is half of a spherical. A hemisphere’s surface area is the total area covered by its surface.

Curved Surface Area of a Hemisphere:

The curved surface area of a hemisphere\( = \frac{1}{2}\left( {{\rm{the}}\,{\rm{curved}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{sphere}}} \right)\)

\( = \frac{1}{2}\left( {4\pi {r^2}} \right) = 2\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)

where, \(r \to {\rm{radius}}\,{\rm{of}}\,{\rm{the}}\,{\rm{hemisphere}}\)

Total Surface Area of a Hemisphere:

The total space occupied or covered by the curved surface and the base surface of the hemisphere is defined as the total surface area of the hemisphere. The sum of the areas of a hemisphere’s curved surface and the base surface can be used to compute its total surface area. If the radius of a hemisphere is known, the surface area can be calculated as follows:

The total surface area of a hemisphere\({\rm{ = Curved}}\,{\rm{surface}}\,{\rm{area + Base}}\,{\rm{area}}\)

\( = 2\pi {r^2} + \pi {r^2} = 3\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)

where \(r \to {\rm{radius}}\,{\rm{of}}\,{\rm{the}}\,{\rm{hemisphere}}\)

Solved Examples

Q.1. Find the total surface area of a cube whose length is \({\rm{12}}\,{\rm{cm}}{\rm{.}}\)
Ans: From the given \({\rm{a = 12}}\,{\rm{cm}}\)
The total surface area \(({\rm{TSA}}) = 6{a^2},\) where \(“a”\) is the length of the side.
\( = 6 \times {12^2}\)
\( = 6 \times 144 = 864\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the total surface area \(({\rm{TSA}})\) of a cube is \(864\;{\rm{c}}{{\rm{m}}^2}\)

Q.2. Find the area of four walls of a room whose length\( = 3.5\,{\rm{m,}}\) breadth\( = 2.5\,{\rm{m}}\) and height \( = 3\,{\rm{m}}{\rm{.}}\)
Answer:
Given: length \((l) = 3.5\,{\rm{m,}}\) breadth\((b) = 2.5\;{\rm{m}}\) and height \((h) = 3\,{\rm{m}}{\rm{.}}\)
We know the area of four walls the is same as the lateral surface area of a cuboid.
Lateral Surface Area of a cuboid \(({\rm{LSA}}) = 2h(l + b)\,{\rm{sq}}{\rm{.units}}\)
\( = 2 \times 3(3.5 + 2.5)\)
\( = 6(6)\)
\( = 36\;{{\rm{m}}^2}\)
Hence, the area of four walls of a room is \(36\;{{\rm{m}}^2}.\)

Q.3. Find the surface area of a spherical ball having a radius of \(7\;{\rm{cm}} \cdot ({\rm{Given}}\,\pi = 3.14)\)
Ans: From the given, \(r = 7\,{\rm{cm}}\)
The surface area of a spherical ball\( = 4\pi {r^2}\)
\( = 4 \times 3.14 \times 7 \times 7\)
\( = 615.44\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the surface area of a spherical ball \(615.44\;{\rm{c}}{{\rm{m}}^2}\)

Q.4. Find the lateral surface area of a right cone if the radius is \(3\;{\rm{cm}}\) and the slant height is \(4\;{\rm{cm}} \cdot ({\rm{Given}}\,\pi = 3.14)\)
Ans: Given: Radius \(r = 3\;{\rm{cm,}}\) Slant height \(l = 4\;{\rm{cm}}\)
Curved Surface Area of a Cone \(({\rm{CSA}}) = \pi rl\,{\rm{sq}}{\rm{.units}}\)
\( = 3.14 \times 3 \times 4\)
\( = 37.68\;{\rm{c}}{{\rm{m}}^2}\)
Hence, the obtained answer is \(37.68\,{\rm{c}}{{\rm{m}}^{\rm{2}}}{\rm{.}}\)

Q.5. Sonu has given a cylinder of surface area \(1728\pi \,{\rm{c}}{{\rm{m}}^2}\) Find the height of the cylinder if the radius of the base of the circle is \({\rm{24}}\,{\rm{cm}}{\rm{.}}\)
Ans: Given: \({\rm{TSA}} = 1728\pi ,r = 24\;{\rm{cm}}\)
Total Surface Area of a Cylinder \({\rm{(TSA)}} = 2\pi r(h + r)\,{\rm{c}}{{\rm{m}}^2}\)
\( \Rightarrow 1728\pi = 2 \times 3.14 \times 24(h + 24)\)
\( \Rightarrow 1728\pi = 2 \times \pi \times 24(h + 24)\)
\( \Rightarrow 1728\pi = 48\pi (h + 24)\)
\( \Rightarrow \frac{{1728}}{{48}} = (h + 24)\)
\( \Rightarrow 36 = h + 24\)
\( \Rightarrow h = 36 – 24\)
\( \Rightarrow h = 12\)
Hence, the height of the cylinder is \(12\,{\rm{cm}}{\rm{.}}\)

Summary

The total area occupied by an object’s surfaces is referred to as its surface area. The area of the surface is divided into two categories are Curved Surface Area (CSA) or Lateral surface Area (LSA), Total Surface Area (TSA). Curved surface area, also known as lateral surface area, is a type of surface area that is curved.

The total surface area is made up of the lateral surface area, as well as the base and top sections. This article contains the surface areas of the three-dimensional shapes. It helps a lot while solving the problems based on it quickly.

Frequently Asked Questions

We have provided some frequently asked questions about surface area formulas here:

Q.1. What is the formula of the curved surface area of the cylinder?
Ans: Curved surface area of a cylinder \(({\rm{CSA}}) = 2\pi rh\,{\rm{sq}}{\rm{.units}}\)

Q.2. What is the formula for the surface area of a cube?
Ans: The formulas to find the surface area of a cube is:
The lateral surface area of a cube \(({\rm{LSA}}) = 4{a^2},\) where \(“a”\) is the length of the side.
The total surface area \(({\rm{TSA}}) = 6{a^2},\) where \(“a”\) is the length of the side.

Q.3. What is the formula for the surface area of a hemisphere?
Ans: The formulas for the surface area of a hemisphere are:
Curved Surface Area of a Hemisphere\( = \frac{1}{2}\left( {4\pi {r^2}} \right) = 2\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)
The total surface area of a hemisphere\({\rm{ = Curved}}\,{\rm{surface}}\,{\rm{area + Base}}\,{\rm{area}}\)
The total surface area of a hemisphere\( = 2\pi {r^2} + \pi {r^2} = 3\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)
where \(r \to {\rm{Radius}}\,{\rm{of}}\,{\rm{the}}\,{\rm{hemisphere}}{\rm{.}}\)

Q.4. What is the formula for the surface area of a sphere?
Ans: We know that the \({\rm{Total}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{sphere = curved}}\,{\rm{surface}}\,{\rm{area}}\,{\rm{of}}\,{\rm{a}}\,{\rm{sphere}}\)
The surface area of a cylinder\( = 4\pi {r^2}\,{\rm{sq}}{\rm{.units}}\)

Q.5. What is the formula of the surface area of cuboids?
Ans: Formulas to find the surface areas of a cuboid are:
The lateral surface area of a cuboid \(({\rm{LSA}}) = 2h(l + b)\,{\rm{sq}}{\rm{.units}}\)
The total surface area of a cuboid \({\rm{(TSA) = 2(}}lh + lb + bh{\rm{)}}\,{\rm{sq}}{\rm{. units}}\)

Q.6. What is the formula of the curved surface area of the cone?
Ans: Curved surface area of a cone \(({\rm{CSA}}) = \pi rl\,{\rm{sq}}{\rm{.units}}\)
where, \(r⟶\)radius of the cone, \(l⟶\)slant height of a cone

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