Collegedunia Team Content Curator
Content Curator
Surface area of a cone is equal to the sum of whole curved surface area and the area of its circular base. When a right-angled triangle is rotated keeping the side of right angle as the central axis, a cone is formed. Thus a cone is a three-dimensional shape having a circular base. The height of the cone is the distance between the center of the base and the tallest part of the cone or its vertex. The formula for the surface area of cone is given by,
Total Surface area of cone = πr (l + r)
Curved Surface area of cone = πrl
Where,
- r is the base radius
- l is the slanting height
Also Check: Volume of Right Circular Cone
| Table of Content |
What is a Cone?
[Click Here for Sample Questions]Cones have a circular base and resemble pyramids. It's a three-dimensional geometry with a smooth base that tapers to a point at the top called the vertex. We analyze a right circular cone with a circular base when studying how to find the surface area of cones. A cone can be thought of as a right-angled triangle rotated around one of its vertices.
Also Read: Conic Sections
What is Surface Area?
[Click Here for Sample Questions]Consider unfolding a shape, or flattening it out, and then calculating the area of each side when calculating the surface area of a 3-D shape. The surface area is calculated by adding all of these areas together. Surface areas are measured in square units such as square meters, square centimeters, or equivalents.
- Curved surface area of a cone covers only the part that is curved.
- Total surface area of a cone is the sum of the curved surface area and the base circle area.
Surface Area of a Cone Formula
[Click Here for Sample Questions]The circumference of the cone's base=2πr, where r is the radius of the circular base's base. As a result,
Curved surface area of cone (CSA) =\(\frac{1}{2} \times l \times 2 \pi r = \pi r l\)
We know that a cone is made by rotating a right-angled triangle. Pythagoras Theorem helps us find the length of the slant side in a right-angled triangle using the formula:
\(l^2 = r^2 + h^2\)
The circular base's surface area = π × r2
Area of the curved surface = πrl
The total curved surface area plus the area of the circular base equals the overall surface area of the cone.
As a result, a Cone's total Surface Area = πr2 + πrl
By taking r as common, we have,
Total Surface area of cone (TSA) = πr (l + r)
Where,
- I = Slant height
- r = The base circle's radius
- SA = Total surface area
- h = Height of cone
Surface Area of Cone in Terms of Radius & Height
The CSA formula for a right circular cone can also be written as follows:
The area of a cone's curved surface is equal to the area of a sector with a radius equal to the slant height ‘l’.
Thus, by Pythagoras theorem, l2 = r2 + h2
l = \(\sqrt{r^2 + h^2}\)
CSA of Cone = \(\pi r \sqrt{(r^2 + h^2)}\)
The TSA formula for a right circular cone can be written as follows:
Total surface area of a cone = Area of a circular base + Curved surface area of a cone (sector's area).
A cone's total surface area = πr2 + πrl
Therefore,
TSA of Cone = πr2 + πr√(r2 + h2)
Also Read: Angle between Two Planes
Surface Area of Cone Derivation
[Click Here for Sample Questions]Consider a cone with a height of "h," a base radius of "r," and a slant height of "l." To find out, we cut the cone open from the center, which looks like a circle sector (a planar shape).
Using the formula, find the area of the sector to get the curved surface area of the cone.
Sector area (in terms of arc length) = (arc length × radius)/ 2 = ((2πr) × l)/2 = πrl.
\(\therefore\) A cone's curved surface area, S = πrl (units)2
The area of a cone's base plus the curved surface area equals the total surface area of the cone.
⇒ The cone's total surface area = πr2 + πrl = πr (r + l).
\(\therefore\) The cone's entire surface area, T = πr (r + l) (units)2
Things to Remember
- Total Surface area of a cone is equal to the sum of whole curved surface area and the area of its circular base.
- The formula for TSA of cone is πr (l + r).
- The formula for CSA of cone is πrl.
- CSA of cone in terms of radius and height is \(\pi r \sqrt{(r^2 + h^2)}\)
- TSA of cone in terms of radius and height is πr2 + πr√(r2 + h2).
Read More:
Sample Questions
Ques: Calculate the area of a cone with a radius of 14 cm and a height of 6 cm. (3 marks)
Solution: Given,
r = 14 centimetres
H = 6 centimetres
As a result, the cone's slant height will be,
\(l = \sqrt{r^2 + h^2} = \sqrt{14^2 + 6^2}\)
l = 15.23 cm
Now, the cone's total surface area = πr (l + r)
\(\frac{22}{7} \times 14 \times (15.23 + 14)\)
= 44×29.23
= 1286.12 square centimeters
As a result, the total surface area is 1286.12 square centimeters.
Ques: Jane spent the weekend camping. She notices a tent with a right circular cone form at the campsite and estimates that the height of the tent is three times the radius (r). Calculate the tent's approximate volume in terms of its radius. (2 marks)
Solution: Given:
The radius, r, is 3 inches, and the height, h, is 4 inches.
Slant height required, s =?
(Slant height)2 = (radius)2 + (height)2
s2 = r2 + h2 = 32 + 42 = 9 + 16
⇒ s2 = 25
⇒ s = \(\sqrt{25}\)
⇒ s = 5
The slant height is 5 inches.
Ques: What is the surface area of a cone with a bottom area of 9π and a height of 4? (2 marks)
Solution: Correct answer 24π.
The radius is determined by the area of the cone's bottom,
\(A = \pi r^2 , r =\sqrt{\frac{A}{\pi}} = \sqrt {\frac{9 \pi}{\pi }} = 3\)
Because the cone's height is 4, the Pythagorean Theorem will yield the slant height,
\(\sqrt{3^2 + 4^2} =5\)
The area of the cone's side is A=πrl=π⋅3⋅5=15π, and when that is added to the 9π stated as the circle's size, the surface area is 24π.
Ques: A cone has a total surface area of 375 square inches. What is the base diameter of the cone if its slant height is four times the radius? (2 marks)
Use π = 3.
Solution: A cone's total surface area = πrl + πr2 = 375 inch2
Slant height:
l = 4 × radius = 4r
Substitute
l = 4r and π = 3
r × 3 × 4 r + 3 × r2 = 375
12r2 + 3r2 = 375
15r2 = 375
r2 = 25
r = \(\sqrt{25}\)
r = 5
As a result, the cone's base radius is 5 inches.
And the cone's base diameter = 2 × radius = 2 × 5 = 10 inch.
Ques: What is the area of a cone's surface? (2 marks)
Ans. The area occupied by a cone's surface in three-dimensional space is referred to as its surface area. It is equal to the sum of the cone's lateral and base areas.
Ques: How can you figure out how big a cone's total surface area is? (2 marks)
Ans. The overall surface area of a cone is equal to the sum of the curved surface and the base area. The entire surface area of the cone is calculated using the following formula:
TSA of cone = πr2 + πrl = πr (l + r) square units.
Ques: What is the area of a cone's flat surface? (2 marks)
Ans. The formula for the flat surface of a cone is πr2 square units because the flat surface of a cone is circular. It's also known as a cone's base surface area.
Ques: How do you calculate a cone's slant height? (2 marks)
Ans. The formula s=\(\sqrt{(r^2 + h^2)}\)units are used to compute the slant height of a cone, where "r" is the radius and "h" is the height of a cone.
Ques: What is a cone's lateral surface area? (2 marks)
Ans. The formula LSA =πr\(\sqrt{(r^2 + h^2)}\)square units are used to calculate the lateral surface area of a cone.
Ques: What Are the Different Types of Cones? (2 marks)
Ans. The right circular cone and the oblique cone are the two forms of cones:
- The base and axis of a right circular cone form a right angle. In geometry, this is the most frequent cone.
- The vertex of an oblique cone is not immediately above the center of the circular base, although it has a circular base.
Comments