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Surface Area Of Cone: Formula To Find Lateral, Curved, Total Surface Area

Surface Area Of Cone

Cone is a three-dimensional structure having a circular base where a set of line segments, connecting all of the points on the base to a common point known as apex. There is a predefined formula for the calculation of Surface Area Of Cone.

The cone formula includes curved surface area, lateral surface area and total surface area of a cone. Students can use these formulas to solve any problem related to area of the cone. These all formulas are well mentioned on the page of recruitmentresult.com.

A cone is a set of non-congruent circular discs that are stacked on one another such that ratio of the radius of adjacent discs remains constant. In order to calculate Surface Area Of Cone, aspirants are advised to scroll down this page.

Surface Area Of Cone

Surface Area Of Cone

Curved surface area of a cone = πrl

Total surface area of a cone = πr (l+r)

Where, r is the base radius, h is the height and l is the slant height of the cone.

Also Check Here: Surface Area of Cuboid Formula

Derivation

For calculate the curved surface area and total surface area of a cone, we divide it into a circular base and the top slanted part. The area of the slanted part gives you the curved surface area. Total surface area is the sum of this circular base and curved surface areas.

Derivation

Area of the circular base:

The base is a simple circle and we know that area of a circle is given as:

Area of a circle=πr²

Where r is the base radius of the cone

Area of the curved surface:

Now if open the curved top and cut into small pieces, so that each cut portion is a small triangle, whose height is the slant height l of the cone.

Curved Surface Area Of A Cone

Curved Surface Area Of A Cone

Now if open the curved top and cut into small pieces, so that each cut portion is a small triangle, whose height is the slant height l of the cone.

Now the area of each triangle =1/2× base of each triangle × l

Area of the curved surface = sum of the areas of all the triangles

From the figure, we know that, the curved surface is equivalent to the perimeter of the base of the cone.

The circumference of the base of the cone = 2πr

Area of the curved surface = 

Area of the curved surface= πrl

Total Surface Area of a Cone = Area of the circular base + Area of the curved surface

Total Surface Area of a Coneπr2 + πrl

Total surface area of a cone = πr(l+r)

Know Here With Example: How To Find Area Of Right Angled Triangle?

Solved Examples

Question 1: A cone has a radius of 3cm and height of 5cm, find total surface area of the cone.

Solution: 
To begin with we need to find slant height of the cone, which is determined by using Pythagoras, since the cross section is a right triangle.

l2 = h2 + r2
l2 = 52 + 32
l2 = 25 + 9
l = √(34)
l = 5.83 cm

And the total surface area of the cone is:
SA = πr2 + πrl
SA = π · r · (r + l)
SA = π · 3 · (3 + 5.83)
SA = 83.17 cm2

Therefore, the total surface area of the cone is 83.17cm2

Question 2: The total surface area of a cone is 375 square inches. If its slant height is four times the radius, then what is the base diameter of the cone? Use π = 3.
Solution: 
The total surface area of a cone = πrl + πr2 = 375 inch2
Slant height: l = 4 × radius = 4r

Substitute l = 4r and π = 3
3 × r × 4 r + 3 × r2 = 375
12r2 + 3r2 = 375
15r2 = 375
r2 = 25
r = 25
r = 5

So the base radius of the cone is 5 inch.
And the base diameter of the cone = 2 × radius = 2 × 5 = 10 inch.

Get All Topics Aptitude Test Formulas: Aptitude Formulas

Question 3: What is the total surface area of a cone if its radius = 4cm and height = 3 cm.
Solution: 
As mentioned earlier the formula for the surface area of a cone is given by:
SA = πr2 + πrl
SA =
πr(r + l)

As in the previous example the slant can be determined using Pythagoras:
l2 = h2 + r2
l2 = 32 + 42
l2 = 9 + 16
l = 5

Insert l = 5 we will get:
SA = πr(r + l)
SA = 3.14 · 4 · (4+5)
SA = 113.04 cm2

Question 4: The slant height of a cone is 20cm. the diameter of the base is 15cm. Find the curved surface area of cone.
Solution:
Given that,
Slant height: l = 20cm
Diameter: d = 15cm
Radius: r = d/2 = 15/2 = 7.5cm

Curved surface area = πrl
CSA = πrl
CSA =π · 7.5 · 20
CSA =471.24 cm2

Get Easiest Methods: Ways to Improve Math Skills

Note:

Dear students, to find the Surface Area Of Cone, you have to use these formulas given on the above section of the page. We have provided some examples also through which you can take help to understand the concept of cone.

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