**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**

**Curved surface area of a cone = ****π**rl

**Total surface area of a cone = ****π**r (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.

**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**

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 = 2****πr**

∴ **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 Cone** = **πr ^{2 }**+

**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.

l^{2} = h^{2} + r^{2}

l^{2} = 5^{2} + 3^{2}

l^{2} = 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 cm

^{2}

Therefore, the **total surface area of the cone is 83.17cm ^{2}**

**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 = **π**rl + **π**r^{2} = 375 inch^{2}

Slant height: l = 4 × radius = 4r

Substitute l = 4r and **π** = 3

3 × r × 4 r + 3 × r^{2} = 375

12r^{2} + 3r^{2} = 375

15r^{2} = 375

r^{2} = 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 = ****πr ^{2} + **

**πrl**

SA =

SA =

**πr(r + l)**

As in the previous example the slant can be determined using Pythagoras:

l

^{2}= h

^{2}+ r

^{2}

l

^{2}= 3

^{2}+ 4

^{2}

l

^{2}= 9 + 16

l = 5

Insert l = 5 we will get:

**SA = πr(r + l)
**SA = 3.14 · 4 · (4+5)

SA = 113.04 cm

^{2}

**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 =

**π**rl

CSA =

**π**· 7.5 · 20

CSA =471.24 cm

^{2}

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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