Introduction
The formula for the surface area of solid shapes in geometry is a mathematical method to calculate the total area occupied by all of the surfaces of any three-dimensional object. Geometric surface area formulas discuss the lateral surface and the overall surface areas of various geometric solid shapes such as cubes, rectangular prisms, cones, cylinders and other shapes.
As a refresher, the surface area of an object is the sum of the areas of all of its faces, or the exterior surfaces of a three-dimensional solid.
It is calculated in square units. In addition, the combined area of the sides of a shape, excluding the area of its base and top, is referred to as the lateral surface area.
Formulas to calculate the Surface Areas of Various Solids
There are two facets to the formulas that are applied to calculate the surface area of solids:
- Formula to find the curved surface area or for the lateral surface area.
- Formula to find the total surface area (that includes the base or bases according to the solid).
The table provided depicts the formulas to help in calculating the surface area for the corresponding 3-D shapes.
Shape | Lateral Surface Area (LSA) | Total Surface Area (TSA) |
|---|---|---|
Cuboid (rectangular prism) | 2h(l + b) | 2(lb + bh + lh) |
Cube | 4a\(^2\) | 6a\(^2\) |
Right Prism | Base perimeter × Height | LSA + 2 (area of one end) |
Right Circular Cylinder | 2πrh | 2πr(r + h) |
Right Pyramid | (\(\frac{1}{2}\)) Perimeter of base x Slant Height | LSA + Area of Base |
Right Circular Cone | πrl | πr(l + r) |
Solid Sphere | 4πr\(^2\) | 4πr\(^2\) |
Hemisphere | 3πr\(^2\) | 3πr\(^2\) |
Rapid Recall
Solved Examples
Example 1:
The slant height of a cone is 18 centimeters, and its radius is 15 cm. Calculate the total surface area of the cone.
Solution:
As stated in the question,
Radius of the cone = 15 centimeters
Slant height of the cone = 18 centimeters
The formula for the total surface area of a cone = πr(r + l)
= 3.14 × 15 × (15 + 18) [Substitute the values]
= 3.14 × 15 × (33) [Apply PEMDAS rule]
= 1554.3 cm\(^2\)
Hence, the surface area of the cone is 1554.3 square centimeters.
Example 2:
A soccer ball has a radius of 20 centimeters. Find the area of the curved surface of the ball. (Use π = 3.14)
Solution:
As stated in the question,
Radius of the soccer ball = 20 centimeters
The curved surface area of sphere = 4πr\(^2\)
= 4π(20)\(^2\) [Substitute the value]
= 4π(400) [Find the square of the radius]
= 4 x 3.14 x (400) [use π = 3.14]
= 5024 cm\(^2\)
Hence, the curved surface area of the soccer ball is 5024 square centimeters.
Example 3:
Calculate the total surface area of a water tank with a radius of 31 inches and a height of 67 inches. (Use π = 3.14)
Solution:
As stated in the question,
Radius of the water tank = 31 inches
Height of the water tank = 67 inches
The total surface area formula of the water tank that is in the shape of a cylinder = 2πr(h + r)
= 2π(31)(67 + 31) [Substitute the values]
= 2π(31)(98) [Apply PEMDAS rule]
= 2 x 3.14 x 31 x 98 [use π = 3.14]
= 19078.64 in\(^2\)
Hence, the total surface area of the water tank is 19078.64 square inches.
The curved surface area of a hemisphere is half of the curved surface area of a sphere.
The formula that helps us find the area covered by the surface of a cylindrical shape is 2πr(h + r), in which ‘r’ is the radius of the circular base of the cylinder, and ‘h’ is its height.
The entire area that the circular cone surface covers is the surface area formula for a right circular cone. It is written as πrL, where ‘r’ denotes the circumference of the cone and ‘l’ is its length.