Table of Contents
What is a Cone?
A cone is a three dimensional solid shape that has a circular base extending in a circular fashion towards a point called the apex or vertex of the cone. The perpendicular distance from the center of the circular base to the apex is known as the height of the cone and the distance from any point on circumference of the base to the apex is known as the slant height of the cone. The radius of a cone is the radius of its circular base.
Here, h represents the height, l represents the slant height and r represents the radius of the cone.
Base Area of a Cone
As mentioned earlier, the base of a cone is circular in shape. So the base area of a cone can be calculated by using the formula for the area of a circle.
Area of circle is given by the formula:
Area = 𝛑r\(^2\)
So the base area of a cone, A = 𝛑r\(^2\)
Where r is the radius of the cone and 𝛑 is a math constant whose value is \(\frac{22}{7}\) or 3.14.
Solved Examples
Example 1: Find the base area of a cone having a base radius of 5 centimeters.[Take π = \(\frac{22}{7}\)]
Solution:
As stated: Base radius of the cone, r = 5 cm
Base area of a cone, A = πr\(^2\) [Formula for base area of cone]
= \(\frac{22}{7}\times 5^2\) [Substitute r as 5]
= \(\frac{22}{7}\times 25\) [Square of number]
= 78.57 cm\(^2\) [Simplify]
Therefore, the base area of the cone is 78.57 square centimeters.
Example 2: The surface area of a cone is 28.28 square inches. Find the radius of the cone.
Solution:
Surface area of the cone = 28.28 in\(^2\)
Base area of a cone, A = π r\(^2\) [Formula for base area of cone]
28.28 = \(\frac{22}{7}\) × r\(^2\) [Substitute the given value]
28.28 × \(\frac{7}{22}\) = r\(^2\) [Multiply both sides by 722]
8.99 = r\(^2\) [Simplify]
\(\sqrt{8.99}\) = r [Positive square root on both sides]
2.99 = r [Square root]
r = 2.99 in
Therefore, the radius of the cone is 2.99 inches.
Example 3: Find the base area of conical funnel with base diameter equal to 14 inches.
Solution:
As stated in question:
Base diameter of the cone = 14 in
Radius is half times the diameter, so,
Base radius of the cone = \(\frac{14}{2}\) = 7 in
Base area of the funnel = π r\(^2\) [Formula for the base area of cone]
= \(\frac{22}{7} \times 7^2\) [Substituting r as 7]
= \(\frac{22}{7} \times 49\) [Square of number]
= 154 in\(^2\) [Simplify]
Therefore the base area of the funnel is 154 square inches.
Lateral surface area of a solid refers to the total area of all its surfaces except the top and bottom faces, which are also referred to as the bases of the three dimensional solid.
The measurement that depicts the area covered by the circular base of a cone is called the base area of the cone.
The height of a cone is the distance from the center of the base to the apex of the cone whereas the slant height of a cone is the distance from any point on the circumference of the base to the apex of the cone. So the height and slant height of a cone are not the same.