The surface area of a three-dimensional object is the total area of all of its surfaces. Surface area is important to know situations where we want to wrap something, paint something, and eventually while building things to get the best possible design.
|1.||What Is Surface Area?|
|2.||Surface Area Formulas|
|3.||FAQ's on Surface Area|
What Is Surface Area?
The total area occupied by the surfaces of an object is called its surface area. The surface area is classified into two categories:
- Curved surface area or Lateral surface area
- Total surface area
Let us learn about the general surface area formulas of various shapes.
Surface Area Formulas
The area of the whole of anything, whether it's an object or a surface, is the sum of the area of its constituent parts. We now know that the surface area of a three-dimensional object is the total area of all of its surfaces. In this section, we will learn about the various formulas used to calculate the surface area of different objects.
Surface Area of Cube
The surface area of the cube is the total area covered by all six faces of the cube. The general formula of the surface area of a cube is given as:
The total surface area of the cube will be the sum of the area of the base and the area of vertical surfaces of the cube. There are a total of 6 surfaces hence, the total surface area = 6s2
The lateral surface area of a cube is the sum of areas of all side faces of the cube. There are 4 side faces so the sum of areas of all 4 side faces of a cube is its lateral area. LSA = 4a2 where "a" is the side length.
Surface Area of Cuboid
The surface area of cuboid can be explained in terms of two different categories of area, i.e., lateral surface area and the total surface area. The total surface area of the cuboid is obtained by adding the area of all the 6 faces whereas the lateral surface area of the cuboid is found by finding the area of each face excluding the base and the top. The total surface area and lateral surface area can be expressed in terms of length (l), breadth(b), and height of cuboid(h) as:
- Total Surface Area of Cuboid, S = 2 (lb + bh + lh) units2
- Lateral Surface Area of Cuboid, L = 2h (l + b) units2
Surface Area of Cone
The surface area of a cone is the amount of area occupied by the surface of a cone. A cone is a 3-D shape that has a circular base. This means the base is made up of a radius and diameter. As a cone has a curved surface, thus we can express its curved surface area as well as total surface area. If the radius of the base of the cone is "r" and the slant height of the cone is "l", the surface area of a cone is given as:
- Total Surface Area, T = πr(r + l)
- Curved Surface Area, S = πrl
Surface Area of Cylinder
A cylinder is a 3-D solid object which consists of two circular bases connected with a curved face. As a cylinder has a curved surface, thus we can express its curved surface area as well as total surface area. If the radius of the base of the cylinder is "r" and the height of the cylinder is "h", the surface area of a cylinder is given as:
- Total Surface Area, T = 2πr(h + r)
- Curved Surface Area, S = 2πrh
Surface Area of Sphere
A sphere is a three-dimensional solid object which has a round structure, like a circle. The area covered by the outer surface of the sphere is known as the surface area of a sphere. The surface area of a sphere is the total area of the faces surrounding it. The surface area of a sphere is given in square units.
The surface area of a sphere is equal to the lateral surface area of a cylinder. Hence, the relation between the surface area of a sphere and lateral surface area of a cylinder is given as:
Surface Area of Sphere = Lateral Surface Area of Cylinder
⇒ The surface area of Sphere = 2πrh
If a diameter of sphere = 2r
Then the surface area of sphere is 2πrh = 2πr(2r) = 4πr2 square units.
Surface Area of Hemisphere
Hemisphere is a three-dimensional shape, obtained when a sphere is cut along a plane passing through the center of the sphere. In other words, a hemisphere is half of a sphere. The surface area of a hemisphere is the total area its surface covers. It can be classified into two categories:
- The curved surface area of a hemisphere(CSA) = ½ (curved surface area of a sphere) = ½ (4 π r2) = 2 π r2 , where "r" is the radius of the hemisphere.
- The total surface area of a hemisphere(TSA) = curved surface area + Base Area = 2 π r2 + π r2 = 3 π r2 , where r is the radius of the hemisphere.
Surface Area of Prism
There are two types of areas we read about, first, the lateral surface area of the prism, and second, the total surface area of the prism. Let us learn in detail.
The lateral area of a prism is the sum of the areas of all its lateral faces whereas the total surface area of a prism is the sum of its lateral area and area of its bases.
The lateral surface area of prism = base perimeter × height
The total surface area of a prism = Lateral surface area of prism + area of the two bases = (2 × Base Area) + Lateral surface area or (2 × Base Area) + (Base perimeter × height).
There are seven types of prisms based on the shape of the bases of prisms. The bases of different types of prisms are different so are the formulas to determine the surface area of the prism. See the table below to understand this concept behind the surface area of various prism:
|Shape||Base||Surface Area of Prism = (2 × Base Area) + (Base perimeter × height)|
|Triangular Prism||Triangular||Surface area of triangular prism = bh + (s1 + s2 + b)H|
|Square Prism||Square||Surface area of square prism = 2a2 + 4ah|
|Rectangular Prism||Rectangular||Surface area of rectangular prism = 2(lb + bh + lh)|
|Trapezoidal Prism||Trapezoidal||Surface area of trapezoidal prism = h (b + d) + l (a + b + c + d)|
|Pentagonal Prism||Pentagonal||Surface area of pentagonal prism = 5ab + 5bh|
|Hexagonal Prism||Hexagonal||Surface area of hexagonal prism = 6b(a + h)
Surface area of regular hexagonal prism = 6ah + 3√3a2
|Octagonal Prism||Octagonal||Surface area of octagonal prism = 4a2 (1 + √2) + 8aH|
Solved Examples on Surface Area
Example 1: While painting a cylindrical tank of radius 3.5 yd and height 6 yd, Ryan thought of painting the outer surface. If the cost of the painting is $5 per yd2, what will be the total cost of the painting?
We know that total Surface Area = Curved Surface Area + area of the top and bottom faces
=2πrh + 2πr2
=2πr(r + h)
=2 × 22/7 × 3.5×(3.5 + 6)
=22 × 9.5
Cost of painting at $5 per yd2 = 209 × 5 = $1045
The cost of the painting is $1045.
Example 2: Dylan observed the cone of the ice cream he was eating. To understand the concept and shape of a cone, his father suggested that he would help him find the total surface area of the cone. Given that the radius = 4 inches and height = 7 inches. What would be the surface area of the ice cream cone?
Given: Radius = 4 inches and height =7 inches
The surface area of cone = πrh + πr2
=3.14 × 4 × 7 + 3.14 × 42
=87.92 + 50.24
∴The surface area of cone will be 138.16 inches2
Example 3: While playing with a Rubik's cube of side length 3 inches, Antonia closely observed the shape that the shape resembled a cube that she learns about while learning shapes in school, she thought of calculating its surface area to get a practical idea. What would be the surface area of the cube?
Solution: Given side length of cube = 3 inches
The surface area of cube = 6s2
s = 3 inches
On substituting values in the formula.
= 6 (3)2
= 6 (9)
= 54 inches2
∴The surface area of a cube will be 54 inches2
FAQ's on Surface Area
Is Surface Area the Same as Area?
The main difference is that surface area is the area of all the constituent parts of 3D shapes such as a sphere, cylinder, etc. whereas area is the measurement of the size of a plane surface, i.e. a 2D shape such as triangle, square, etc.
How Do You Find the Surface Area of a Solid Shape?
Let us take an example of a cuboid-shaped solid of length 8 inches, breadth(width) 6 inches, and height 5 inches. What will be the surface area of the cuboid?
Answer: Given, a = 8 inches, b = 6 inches, h = 5 inches
The total surface area=2(ab+ah+bh)=2×[(8×6)+(5×6)+(8×5)]=236 sq.inches
What Is the Surface Area of a Circle?
The surface area of a circle is the total area covered by the boundary of a circle, i.e., circumference. The area of a circle with radius "r" is given as πr2
What is the Relation Between Volume and Surface Area of a Cuboid?
The volume of a cuboid is expressed as the product of the height of the cuboid and the area of one surface of the cuboid. It is given as V = lbh, where "l" is length, "b" is the breadth, and "h" is height. Here "lb" is area of the rectangular face of cuboid.
What is the Surface Area of a Cone?
The measure of the area occupied by the surface of a cone is referred to as the surface area of a cone. It is given as total surface area of cone, T = πr(r + l) and curved surface area of cone, S = πrl. Here "r" is radius of the base of the cone and "l" is slant height of the cone.
What Is the Surface Area of a Cylinder?
The surface area of a cylinder is the total region covered by the surface of the cylindrical shape. The total surface area of a cylinder is given as the sum of lateral surface area and the area of two bases. It is mathematically expressed as 2πr(h+r) and is given in square units, like m2, in2, cm2, yd2, etc.