Surface Area Formulas

The surface area formula is a mathematical solution to find the total area of any three-dimensional object occupied by all of its surfaces. Let us understand in detail about surface area formulas of different 3-D shapes.

What Is Surface Area Formula?

The surface area formula is used to find the sum of all the surface areas of any three-dimensional object. The surface area formula is classified under two categories:

• Curved surface area formula or Lateral surface area formula
• Total surface area formula

Look at the surface area formulas chart below depicting the surface area formulas of respective 3-D shapes.

Let us learn about the general surface area formulas of various shapes in detail.

Surface Area Formulas of Different Shapes

Any three-dimensional object has lateral surfaces and the base surface(s). The Total surface area refers to the sum of both the lateral/curved surface area and the base surface area. In this section, we will learn about the surface area formulas of various 3-D shapes. 

Surface Area Formula 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 formula of the cube will be the sum of the area of vertical surfaces of the cube and the area of the base. The total surface area formula of cube = 6a² where "a" is the side length.
• The lateral surface area formula of a cube is the sum of areas of all lateral side faces of the cube. LSA = 4a² where "a" is the side length.

Surface Area Formula of Cuboid

The total surface area formula of the cuboid is obtained by adding the area of all 6 faces. The total surface area and lateral surface area can be expressed in terms of its dimensions: 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 Formula of Cone

A cone is a 3-D shape that has a circular base with a radius "r" and diameter "d". It has a curved surface, thus we can have its curved surface area formula as well as total surface area formula. 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 of a cone, T = πr(r + l)
• The curved surface area of a cone, S = πrl

Surface Area Formula of Cylinder

A cylinder has a curved surface with two circular bases placed at both ends. 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 of cylinder, T = 2πr(h + r)
• The curved surface area of a cylinder, S = 2πrh

Surface Area Formula of Sphere

A sphere is a three-dimensional solid object with a round structure. The surface area of a sphere is the total area of the faces surrounding it. The surface area formula of the sphere is given as:
⇒ The surface area of Sphere, S = 4πr² square units.

Surface Area Formula of Hemisphere

Hemisphere is half of a sphere. The surface area of a hemisphere is the total area its surface covers. The surface area formula of the hemisphere 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 Formula of Prism

The lateral surface 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 the area of its bases. The surface area formulas of the prism can be given

• 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. Check out prism to understand the concept behind the surface area formulas of various prisms.

Surface Area Formula of Pyramid

If a pyramid has a regular polygon base with altitude passing through the center of the base, then the lateral surface area and total surface area formulas for the pyramid can be given as:

Consider a regular pyramid whose base perimeter is 'P', the base area is 'B', and the slant height (the height of each triangle) is 's'. Then,

• The lateral surface area of pyramid (LSA) = (1/2) Ps
• The total surface area of pyramid (TSA) = LSA + base area = (1/2) Ps + B

Examples Using Surface Area Formula 

Example 1: A cylindrical tank has a radius 4 yd and height 8 yd, using the surface area formula of the cylinder find its surface area. If the cost of the painting cylindrical tank is $6 per yd², what will be the total cost of the painting?

Solution:

We know that total Surface Area formula of cylinder = Curved Surface Area of cylinder + area of the top and bottom faces
=2πrh + 2πr²
=2πr(r + h)
=2 × 22/7 × 4 × (4 + 8)
=301.68 yd²
Cost of painting at $6 per yd² = 301.68 × 6 = $1810.08
The cost of the painting is $1810.08. 

Example 2: Given that the radius of a cone is 6 inches and the slant height of a cone is 9 inches. Using the total surface area formula of the cone calculate the surface area of the cone.

Solution:

Given: Radius = 6 inches and slant height = 9 inches
The total surface area formula of cone = T = πr(r + l)
=3.14 × 6 × (6 + 9)
=282.6 inches²
∴The surface area of cone will be 282.6 inches²

Example 3: Using the surface area formula of the cube find the surface area of the cube whose side is 4 inches.

Solution: Given side length of cube = 4 inches
The surface area formula of cube = 6a²
a = 4 inches
On substituting values in the surface area of a cube formula. 
= 6 (4)²
= 6 (16)
= 96 inches²
∴The surface area of a cube will be 96 inches²