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Surface Area of a Cone

The first step in finding the surface area of a cone is to measure the radius of the circle part of the cone. The next step is to find the area of the circle, or base. The area of a circle is 3.14 times the radius squared (πr2). Now, you will need to find the area of the cone itself. In order to do this, you must measure the side (slant height) of the cone. Make sure you use the same form of measurement as the radius.

You can now use the measurement of the side to find the area of the cone. The formula for the area of a cone is 3.14 times the radius times the side (πrl).

So the surface area of the cone equals the area of the circle plus the area of the cone and the final formula is given by:

SA = πr2 + πrl


 
Where,
r is the radius
h is the height
l is the slant height

The area of the curved (lateral) surface of a cone = πrl
 Note:
A cone does not have uniform (or congruent) cross-sections. (more about conic section here)


Example 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


Example 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.


Example 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

Example 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


Example 5: Height and radius of the cone is 5 yard and 7 yard. Find the lateral surface area of the given cone.
Solution:
Lateral surface area of the cone = πrl

Step 1:
Slant height of the cone:
l2 = h2 + r2
l2 = 72 + 52
l2 = 49 + 25
l = 8.6

Step 2: Lateral surface area:
LSA = πrl
LSA = 3.14 × 7 × 8.6
LSA =189.03 yd2

So, the lateral surface area of the cone = 189.03 squared yard.


Example 6: A circular cone is 15 inches high and the radius of the base is 20 inches What is the lateral surface area of the cone?

Solution:

The lateral surface area of cone is given by:
LSA = π × r × l
LSA =3.14 × 20 × 15
LSA = 942 inch2


Example 7: Find the total surface area of a cone, whose base radius is 3 cm and the perpendicular height is 4 cm.
Solution:
Given that:
r = 3 cm
h = 4 cm

To find the total surface area of the cone, we need slant height of the cone, instead the perpendicular height.
The slant height l can be found by using Pythagoras theorem.

l2 = h2 + r2
l2 = 32 + 42
l2 = 9 + 16
l = 5

The total surface area of the cone is therefore:
SA = πr(r + l)
SA = 3.14 · 3 · (3+5)
SA = 75.36 cm2


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