What is the Surface Area of an Object?
The surface area of a three-dimensional object is the sum of the area of all of its faces. When wrapping, painting, and eventually building things to achieve the accurate and appealing designs, we apply the concept of the surface area of various items in real life.
How to Calculate the Surface Area of an Object?
Calculating the surface area of an object is a mathematical method for calculating the total area occupied by all of the surfaces of an object. To calculate the total surface area of any three-dimensional object, we add the area of all of its faces. There are two types of formulas to find the surface area depending on the type of three-dimensional object:
- Formula for the curved surface area, or, formula for the lateral surface area
- Formula for the total surface area
Look at the chart below that shows the surface area formulas for the corresponding 3-D shapes.
Formula for the Surface Area of a Cube:
The total area occupied by six faces of the cube is known as its surface area. The following is the formula for the surface area of cube:
- Total surface area of a cube, \(s=6a^{2}\) square units, where ‘a’ is the side length.
- Lateral surface area of a cube, \(LSA=4a^{2}\) square units, where ‘a’ stands for side length.
Formula for the surface area of a cuboid:
The total surface area of a cuboid is calculated by adding the areas of each of its six faces. The length (l), width (b), and height (h) are used to express the total surface area and lateral surface area:
- Total surface area of a cuboid, S = 2 \(s=2(lb+bh+lh)\) square units
- Lateral surface area, \(LSA=2h(l+b)\) square units
Formula for the Surface Area of a Cone
A cone is a three-dimensional shape with a circular base and with dimensions as ‘r’, the radius of the base and ‘l’, the slant height of the cone. Since it has a curved surface, we may calculate both the overall surface area and the curved surface area.
- Total surface area of a cone, \(S=\pi r(r+l)\) square units
- Lateral surface area of cone, \(LSA=\pi rl\) square units
Formula for the Surface Area of a Cylinder:
Two circular bases are connected with a curved surface. The surface area of a cylinder is given as follows if the radius of the base is ‘r’ and the height is ‘h’:
- Total surface area of a cylinder, \(S=2\pi r(h+r)\) square units
- Lateral surface area of a cylinder, \(LSA-2\pi rh\) square units
Formula for the Surface Area of a Sphere:
A three-dimensional solid object featuring a fully curved surface without vertices is called a sphere. The surface area of a sphere is calculated as follows:
Surface area of sphere, \(S=4\pi r^{2}\) square units
Formula for the Surface Area of a Hemisphere:
A hemisphere as the name suggests is half of a sphere. The surface area of a hemisphere is the area of the curved surface only, or the curved surface along with the circular base. The radius of the hemisphere is denoted by ‘r’.
- The curved surface area of a hemisphere, \(CSA=2\pi r^{2}\) square units
- Total surface area of a hemisphere, \(TSA=3\pi r^{2}\) square units
Formula for the Surface Area of a Prism:
- The lateral surface area of a prism = base perimeter × height square units
- The total surface area of a prism =(2 × Base Area) + (Base perimeter × height) square units
Formula for the Surface Area of a Pyramid:
Think of a conventional pyramid with a base area of ‘B’, a base perimeter of ‘P’, and a slant height of ‘H’, then,
- The lateral surface area of a pyramid \((LSA)=\left ( \frac{1}{2} \right )PH\) square units
Total surface area of a pyramid \(TSA=\left ( \frac{1}{2} \right )PH+B\) square units
Solved Examples
Example 1: Rosie wanted to paint a glass cube that she bought at a garage sale. Find the area of the cube that Rosie has to paint with each side measuring 6 inches.
Solution:
Details in the question,
Side length of the cube, a= 6 in.
Formula for the surface area of a cube,
\(S=6a^{2}\) [Write the formula]
\(=6(6a)^{2}\) [Substitute the values]
\(=6\times 36\) [Calculate the square]
\(=216in^{2}\) [Multiply]
So, the surface area of the cube Rosie had to paint is 216 square inches.
Example 2: What is the surface area of an ice cream cone if its radius is 7 centimeters and slant height is 9 centimeters?
Solution:
Radius = 7 cm
Slant height = 9 cm
\(S=\pi r(r+l)\) [Write the formula]
\(=3.14\times 7(7+9)\) [Substitute the values]
\(=3.142\times 7(16)\) [Add]
\(=351.904cm^{2}\) [Multiply]
Hence, the surface area of the ice cream cone is 351.904 square centimeters.
Example 3: Can you calculate the surface area of a cylindrical tank with a radius of 50 centimeters and a height of 100 centimeters. What is the cost of painting the cylindrical tank, if the cost of painting it is $0.10 per square centimeter?
Solution:
Use the formula for total surface area formula of a cylinder,
\(S=2\pi r(r+h)\) [Write the formula]
\(=2\times3.142 \times50\times(50+100)\) [Substitute the values]
\(=47124cm^{2}\) [Multiply]
Cost to paint the cylindrical tank at $0.10 per square centimeter
\(=47124\times0.10=$4712.40\)
The cost to paint the cylindrical tank is $ 4712.40.
The main distinction between area and surface area is that the former measures the space occupied by 2D shapes like triangles, squares, and so on, while the latter, the surface area, measures the area of 3D shapes like a sphere, cylinder, and so on.
The surface area of 3D objects is the entire area that is covered by all of its faces, and is expressed in square units. For instance, if we need to determine how much paint is needed to paint a box, the surface area of the box gives us the area that will be painted.
The formula that is applied to obtain the surface area of a rectangular prism is equal to 2(lw + wh + lh), where ‘l’, ‘w’, and ‘h’ are its length, width, and height.