Question

find the slant height and vertical height of the cone with radius 1.4cm and curved surface area 158.4 cm​

Answer 1

Given :-

• Curved surface area of the cone = 158.4cm²
• Radius of the cone = 1.4cm

To find :-

• Slant height
• Vertical height

Let us assume slant height to be 'l' cm

Curved surface area of cone :- πrl

Where r is the radius and 'l' is the slant height

By substituting the values,

$158.4 = 1.4 \times \pi \times l$

$158.4 = 1.4 \times \dfrac{22}{7} \times l$

By reducing to the lowest terms, [cancelling 1.4 and 7]

$158.4 = 4.4 \times l$

Transposing 4.4 to the LHS (Left Hand Side),

$\dfrac{158.4}{4.4} = l$

$36 = l$

Therefore the slant height is 36cm

Let us assume the vertical height to be 'h' cm.

Notice that when the radius and vertical height meeting at the point O, the triangle formed is right angled

Hence by applying the Pythagoras theorem, we can find the vertical height

Pythagoras theorem :-

The Pythagoras theorem states that the base squared (in this case the radius) and the height squared (vertical height) will be equal to the hypotenuse squared (slant height) in a right angled triangle.

By substituting the values,

(base)² + (height)² = (hypotenuse)²

r² + h² = l²

(1.4)² + h² = (36)²

1.96 + h² = 1,296

Transposing 1.96 to the RHS (Right Hand Side),

h² = 1,296 - 1.96

h² = 1,294.04

Transposing the power,

$h = \sqrt{1294.04}$

Vertical height = √1294.04 cm = 35.97cm (approximately)

Some more formulas :-

Curved surface area of a cube = 4a²

Curved surface area of a cuboid = 2h(l+b)

Curved surface area of a cylinder = 2πrh