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The surface area of a three-dimensional object is the total area of all its faces. In real-life we use the concept of surface areas of different objects when we want to wrap something, paint something, and eventually while building things to get the best possible design. Let us learn all about the surface area of 3D shapes in this article.
|1.||What is Surface Area?|
|2.||Surface Area Formulas|
|3.||Types of Surface Areas|
|4.||Surface Area of Prism|
|5.||FAQs on Surface Area|
What is Surface Area?
The total area occupied by the surfaces of an object is called its surface area. In geometry, different 3D shapes have different surface areas which can be easily calculated using the formulas that we will be learning in this article. The surface area is classified into two categories:
- Lateral surface area or Curved surface area
- Total surface area
Let us learn about the general surface area formulas of various shapes.
Surface Area Formulas
There is a different surface area formula for every geometrical shape, but the idea behind all is to get the total area occupied by all the faces of the objects. In this section, we will learn about the various formulas used to calculate the surface area of different objects. The total surface area considers all the faces of the 3D shape including the flat surfaces and the curved surfaces, while the lateral surface area is calculated to find the area occupied by the curved surface of the shape. It does not include the area of the bases. A sphere is one 3D figure which has only one round surface with no flat base.
Observe the table given below to learn the surface area formulas of different 3D shapes.
|3D Shape||Total Surface Area (TSA)||Lateral Surface Area (LSA)/Curved Surface Area|
|Cube||6a2||4a2, where a is the length of each side|
|Cuboid||2 (lw + wh + lh)||2h (l + w), where l, w, and h are the length, width, and height of the cuboid|
|Cone||πr(r + l)||πrl, where r is the radius and l is the slant height of the cone|
|Cylinder||2πr(r + h)||2πrh, where r is the radius and h is the height of the cylinder|
|Sphere||4πr2, where r is the radius of the sphere||Not applicable|
Types of Surface Areas
As we have already discussed that there are two types of surface areas for three-dimensional shapes: total surface area and curved/lateral surface area. The total surface area includes the area of all the faces of the shape while the curved or lateral area includes only the area of the side faces of the shapes. Observe the cylinder given below to understand the difference between total surface area and curved surface area.
Surface Area of Prism
A prism is a 3D solid object made up of two congruent bases which are polygons and congruent lateral faces which are rectangular in shape. There are two types of areas that a prism has - the lateral surface area and the total surface area. 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 = (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. Observe the table given below to understand this concept behind the surface area of various prisms:
|Shape||Base||Surface Area of Prism = (2 × Base Area) + (Base perimeter × height)|
|Triangular Prism||Triangle||Surface area of triangular prism = bh + (s1 + s2 + b)H|
|Square Prism||Square||Surface area of square prism = 2a2 + 4ah|
|Rectangular Prism||Rectangle||Surface area of rectangular prism = 2(lw + wh + lh)|
|Trapezoidal Prism||Trapezoid||Surface area of trapezoidal prism = h (b + d) + l (a + b + c + d)|
|Pentagonal Prism||Pentagon||Surface area of pentagonal prism = 5ab + 5bh|
|Hexagonal Prism||Hexagon||Surface area of hexagonal prism = 6ah + 3√3a2|
|Octagonal Prism||Octagon||Surface area of octagonal prism = 4a2 (1 + √2) + 8aH|
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Surface Area Examples
Example 1: Find the total surface area of a cylinder if its radius is 3.5 units and height is 6 units.
We know that the formula to find the total surface area of a cylinder = 2πr(r + h)
= 2 × 22/7 × 3.5 × (3.5 + 6)
= 2 × 22/7 × 3.5 × (9.5)
= 209 unit2
Therefore, the total surface area of the cylinder is 209 unit2
Example 2: If the radius and slant height of an ice cream cone is 4 inches and 7 inches respectively, what is its surface area?
Given: radius = 4 inches and slant height = 7 inches.
The surface area of cone = πr(r + l)
= π × 4(4 + 7)
= 3.14 × 4 × 11
= 138.16 inches2
∴ The surface area of the cone is 138.16 inches2.
Example 3: What is the surface area of a cube whose each side measures 3 inches?
Solution: Given side length of cube = 3 inches.
The surface area of a cube = 6a2
a = 3 inches (given)
On substituting the values in the formula, we get,
= 6 (3)2
= 6 (9)
= 54 inches2
∴ The surface area of the cube is 54 inches2.
FAQs on Surface Area
What is the Definition of Surface Area?
The surface area is the total area covered by all the faces of a 3D object. For example, if we need to find the quantity of paint required to paint a cube, then the surface on which the paint will be applied is its surface area. It is always measured in square units.
What is the Formula for Surface Area?
The formula to find the surface areas of different geometrical shapes is to add the areas of each of their faces. It can be very tedious to find the area of each face individually, so we have surface area formulas for each of the geometrical figures. Some of the formulas are listed below:
- Total surface area of a cube = 6 × (side)2
- Surface area of sphere = 4πr2
- Total surface area of cone = πr(r + l)
Is Surface Area the Same as Area?
The main difference between surface area and area is that surface area is the area of 3D shapes such as a sphere, cylinder, and so on, whereas, area is the measurement of the space occupied by 2D shapes such as triangles, squares, and so on.
How to Find the Surface Area of a Rectangular Prism?
The surface area of a rectangular prism can be calculated by using the following formula: Surface area of rectangular prism = 2(lw + wh + lh), where l, w, and h are the length, width, and height of the rectangular prism respectively.
What is the Surface Area of a Cube?
A cube is made up of 6 square faces. So, the surface area of a cube is the sum of the areas of all those 6 faces. We know that area of a square = a2, where a is the side length of the square. So, the surface area of a cube with side length a is 6a2.
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. The area of a circle with radius 'r' is, Area of a circle = πr2.
What is the Surface Area of a Cone?
The area occupied by a cone is referred to as the surface area of a cone. The total surface area of a cone is, T = πr(r + l), and the curved surface area of a cone, S = πrl. Here 'r' is the radius of the base and 'l' is the 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(r + h) and is expressed in square units, like m2, in2, cm2, yd2, etc. The curved or the lateral surface area of a cylinder is calculated with the formula, Curved surface area = 2πrh.