**NCERT Class 9 Maths Lab Manual – Form a Cone from a Sector of a Circle**

**Objective**

To form a cone from a sector of a circle and to find the formula for its curved surface area.

**Materials Required**

- A piece of wooden hardboard
- Acrylic sheets
- White paper
- Adhesive tape
- Scissors
- Geometry box
- Marker

**Prerequisite Knowledge**

- Concept of a circle.
- Concept of sector of a circle.
- Concept of a cone.

**Theory**

- For concept of a circle refer to Activity 23.
**Sector of the Circle:**

The region between an arc and the two radii joining the centre to the end points of the arc is called a sector.

Sectors are of two types – minor sector and major sector. Minor sector is the sector of circle, which is less than a semi-circle and major sector is the sector of circle, which is greater than a semi-circle, (see Fig. 28.1)

Area of sector = \(\frac { Arc\quad length }{ Circumference\quad of\quad circle } \times { Area\quad of\quad the\quad circle }\)

**Cone:**A right circular cone is a solid generated by revolving a line segment which passes through a fixed point and which makes a constant angle with a fixed line.

In other words, if a right angled triangle is revolved about one of the two sides forming a right angle, keeping the other sides fixed in position, then the solid so obtained by revolving the line segments is called a right circular cone.

In Fig. 28.2, a right angled ∆OPA on revolving about the segment OP, generates a right circular cone in which ABC is a circle.

**Procedure**

- Take a piece of wooden hardboard of suitable size and by using adhesive, paste a white paper on it.
- From acrylic sheet, cut out a circle of radius l. (see Fig. 28.3)

- Now, cut out a sector having angle θ° from the circle, (see Fig. 28.4)

- To form a cone, bring together both the radii of the sector and by using a adhesive tape, attach the ends and fix it on the hardboard. (see Fig. 28.5)

**Demonstration**

- Radius of the base of cone = r
- Slant height of the cone = Radius of circle = l
- Circumference of the base of cone = Arc length of sector = 2πr
- Now, curved surface area of cone = Area of the sector

= Area of sector = \(\frac { Arc\quad length }{ Circumference\quad of\quad circle } \times { Area\quad of\quad the\quad circle }\)

= \(\frac { 2\pi r }{ 2\pi l } \times \pi { l }^{ 2 }\)

= πrl

**Observation**

By actual measurement,

The slant height (l) of the cone = ………… and radius (r) = …………

∴ Arc length, (l) = ………….

Area of the sector = ………….

curved surface area of the cone = …………

Hence, curved surface area of the cone = Area of the sector

**Result**

We have derived the formula for calculating the curved surface area of cone.

**Applications**

This result is useful in

- estimation of canvas required to make a conical tent.
- estimation of material required to make joker’s cap, ice-cream cone, etc.

**Viva-Voce**

**Question 1.**

What is the sector of a circle?

**Answer:**

The sector of a circle is the portion which is enclosed by two radii and an arc.

**Question 2.**

How will you define a cone?

**Answer:**

A cone is a three dimensional geometrical shape that has one circular base and one vertex.

**Question 3.**

What is the formula for finding the curved surface area of a cone of radius r and slant height l?

**Answer:**

Curved surface area of a cone = πrl

**Question 4.**

Do you know about any formula for finding the area of base of a cone?

**Answer:**

Yes, we know that area of base of a cone can be calculated with help of the formula for finding the area of a circle, i.e. πr².

**Question 5.**

What is the slant height of a cone having radius r and height h?

**Answer:**

Slant height, l = √(h² + r²)

**Suggested Activity**

After forming a cone from the sector of a circle, verify experimentally that curved surface area of cone is equal to the area of the sector.

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