# Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained

**Solution:**

Consider a frustum of a cone with h as height, l as the slant height, r₁ and r₂ as radii of the ends where r₁ > r₂

To Prove:

(i) CSA of the frustum of the cone = πl (r₁ + r₂)

(ii) TSA of the frustum of the cone = πl (r₁ + r₂) + πr₁² + πr₂²

where r₁, r₂, h and l are the radii height and slant height of the frustum of the cone respectively.

Construction:

Extended side BC and AD of the frustum of cone to meet at O

Proof:

The frustum of a cone can be viewed as a difference of two right circular cones OAB and OCD.

Let h₁ and l₁ be the height and slant height of cone OAB and h₂ and l₂ be the height and slant height of cone OCD respectively.

In ΔAPO and ΔDQO

∠APO = ∠DQO = 90° (Since both cones are right circular cones)

∠AOP = ∠DOQ (Common)

Therefore, ΔAPO ∼ ΔDQO (AA criterion of similarity)

AP/DQ = AO/DO = OP/OQ (Corresponding sides of similar triangles are proportional)

⇒ r₁/r₂ = l₁/l₂ = h₁/h₂

⇒ r₁/r₂ = l₁/l₂ or ⇒ r₂/r₁ = l₂/l₁

Subtracting 1 from both sides we get

r₁_{ }/ r₂_{ }- 1 = l₁_{ }/ l₂_{ }- 1

(r₁ - r₂)/r₂ = (l₁ - l₂) / l₂

(r₁ - r₂)/r₂ = l/l₂ [From diagram, l₁ - l₂ = l]

l₂ = lr₂/(r₁ - r₂).... (i)

or r₂_{ }/ r₁ = l₂_{ }/ l₁

Subtracting 1 from both sides we get

r₂_{ }/ r₁ - 1 = l₂_{ }/ l₁ - 1

(r₂ - r₁) / r₁ = (l₂ - l₁) / l₁

(r₁ - r₂) / r₁ = (l₁ - l₂) / l₁

(r₁ - r₂) / r₁_{ }= l_{ }/ l₁

l₁ = lr₁/(r₁ - r₂) .... (ii)

(i) CSA of frustum of cone = CSA of cone OAB - CSA of cone OCD

= πr₁l₁ - πr₂l₂

= π (r₁l₁ - r₂l₂)

= π [(r₁ × lr₁/(r₁ - r₂) - r₂ × lr₂/(r₁ - r₂)] [Using (i) and (ii)]

= π [(lr₁² - lr₂²)/(r₁ - r₂)]

= π [l(r₁² - r₂²)/(r₁ - r₂)]

= π [l(r₁ - r₂)(r₁ + r₂)/(r₁ - r₂)] [Since, a² - b² = (a - b)(a + b)]

= π l(r₁ + r₂)

TSA of frustum of cone = CSA of frustum + Area of lower circular end + Area of top circular end

= πl (r₁ + r₂) + πr₁² + πr₁²

Therefore, CSA of the frustum of the cone = πl (r₁ + r₂)

TSA of the frustum of the cone = πl (r₁+ r₂) + πr₁² + πr₂²

Hence Proved.

**☛ Check: **Class 10 Maths NCERT Solutions Chapter 13

**Video Solution:**

## Derive the formula for the curved surface area and total surface area of the frustum of a cone, given to you in Section 13.5, using the symbols as explained

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.5 Question 6

**Summary:**

The formula for the curved surface area and total surface area of the frustum of a cone are πl(r₁+r₂) and πl(r₁+r₂)+πr₁² +πr₂² respectively where r₁,r₂, h and l are the radii,height and slant height of the frustum of the cone respectively has been derived.

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