# FIND HEIGHT AND SLANT HEIGHT OF THE CONE WHEN SURFACE IS GIVEN

Write the given surface area

(i) Equate the formula to the surface area given.

(ii) Apply the known values, to figure out unknown.

To find lateral surface area and total surface area of cone, we use the formulas given below.

Lateral surface area =  πrl

Total surface area =  πrl +  πr2

= πr(l + r)

l = √r2 + h2

Here r = radius, l = slant height

Find the slant height (h) and slant height (l) of the cone given below. Total surface area is given.

Problem 1 :

Solution :

Surface area of cone = 33π

π r(l + r) = 33π

Applying the value of r, we get

3(l + 3) = 33

Dividing by 3.

l + 3 = 11

Subtracting 3 on both sides.

l = 11 - 3

l = 8

h = √l2 - r2

h = √82 - 32

h = √(64 - 9)

h = √55

So, required height = √55 inches and slant height = 8 inches.

Problem 2 :

Solution :

Surface area of cone = 126π

Diameter = 12 cm

π r(l + r) = 126π

Applying the value of r, we get

6(l + 6) = 126

Dividing by 6.

l + 6 = 21

Subtracting 6 on both sides.

l = 21 - 6

l = 15

h = √l2 - r2

h = √152 - 62

h = √(225 - 36)

h = √189

So, required height = 15 cm and slant height = √189 cm.

Problem 3 :

Solution :

Surface area of cone = π

π r(l + r) = 60π

Applying the value of r, we get

5(l + 5) = 60

Dividing by 5.

l + 5 = 12

Subtracting 5 on both sides.

l = 21 - 5

l = 16

h = √162 - 52

h = √256 - 25

h = √231

So, the required slant height is 16 ft and height is √231 ft.

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