# A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Choose value of π as found appropriate)

**Solution:**

A figure is drawn below to visualize the double cone formed.

In order to find the volume and surface area, we need to find BD or radius of the double cone

From the figure it can be seen that BD ⊥ AC

In ΔABC right-angled at B using Pythagoras theorem,

AC² = AB² + BC²

AC = (3 cm)² + (4 cm)²

= 9 cm² + 16 cm²

= 25 cm²

= 5 cm

Consider ΔABC and ΔBDC

∠ABC = ∠CDB = 90° (BD ⊥ AC)

∠BCA = ∠BCD (common)

By AA criterion of similarity ΔABC ∼ ΔBDC

Therefore,

AB/BD = AC/BC (Corresponding sides of similar triangles are in proportion)

BD = (AB × BC)/AC

= (3 cm × 4 cm)/5 cm

= 12/5 cm

= 2.4 cm

We know that, Volume of the cone = 1/3πr²h

Volume of double cone = Volume of Cone ABB’ + Volume of Cone BCB’

= 1/3 × π(BD)² × AD + 1/3π (BD)² × DC

= 1/3 × π(BD)² [AD + DC]

= 1/3 × π(BD)² × AC

= 1/3 × 3.14 × 2.4 cm × 2.4 cm × 5 cm

= 30.144 cm³

We know that, CSA of frustum of a cone = πrl

CSA of double Cone = CSA of cone ABB’ + CSA of cone BCB’

= π × BD × AB + π × BD × BC

= π × BD [ AB + BC]

= 3.14 × 2.4 cm × [3 cm + 4 cm]

= 3.14 × 2.4 cm × 7 cm

= 52.752 cm²

= 52.75 cm²

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

**Video Solution:**

## A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed

NCERT Solutions for Class 10 Maths Chapter 13 Exercise 13.5 Question 2

**Summary:**

A right triangle, whose sides are 3 cm and 4 cm (other than hypotenuse) is made to revolve about its hypotenuse. The volume and surface area of the double cone so formed are 30.144 cm³ and 52.75 cm² respectively.

**☛ Related Questions:**

- A cistern, internally measuring 150 cm × 120 cm × 110 cm, has 129600 cm3 of water in it. Porous bricks are placed in the water until the cistern is full to the brim. Each brick absorbs one-seventeenth of its own volume of water. How many bricks can be put in without overflowing the water, each brick being 22.5 cm × 7.5 cm × 6.5 cm?
- In one fortnight of a given month, there was a rainfall of 10 cm in a river valley. If the area of the valley is 7280 km², show that the total rainfall was approximately equivalent to the addition to the normal water of three rivers each 1072 km long, 75 m wide and 3 m deep.
- An oil funnel made of tin sheet consists of a 10 cm long cylindrical portion attached to a frustum of a cone. If the total height is 22 cm, diameter of the cylindrical portion is 8 cm and the diameter of the top of the funnel is 18 cm, find the area of the tin sheet required to make the funnel.
- 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.