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Surface Areas and Volumes

Cone

Have you ever wondered how much ice cream is needed to fill an ice cream cone? Doesn’t the cone look like a triangle with a curved surface? So, how is the conical shape formed? And what is its volume and surface area? Let us try to find out answers to these questions.

Cone

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Cone

In geometry, the word cone refers to a pyramid-like structure with a circle-shaped base. It is a three-dimensional figure and has a circular base that tapers to a point called a vertex, apex or top. The shortest distance between the vertex and the base is called height. And, the distance from the vertex to a point on the base is the slant height.

Types of Cone

Right Circular Cone: In this type, the axis makes a right angle from the base.

Oblique Cone: In this type, the axis is non-perpendicular to the base.

Surface Area of a Cone

While studying cones, we generally consider a right circular cone. If you cut a paper cone straight along its side and open it, you can see the shape of paper that forms the surface of the cone. And, when you bring the sides marked P and Q together, you can see the curved portion of the shape forms a circle. Furthermore, if this shape is cut into small pieces along the lines marked in the figure, you can see that the pieces look like a triangle whose height is the slant height, l.

cone

Now, the area of each triangle = 1/2×base of triangle×l.  So, the sum of the areas of many such triangles makes up the area of the sector (fig c). Hence, the curved surface area is

= 1/2 b1l +1/2 b2l + 1/2 b3l +………. = 1/2 l (b1 + b2+ b3 + …) = 1/2 × l× (length of the whole curved boundary)

 

Also, the entire curved portion makes the perimeter of the base of cones. The circumference of the base = 2πr, where r is the base of the radius of the base.

So, the curved surface area = 1/2 *l* 2πr = πrl or simply

Curved Surface Area of Cone = πrl

where r is the radius of the base and l is its slant height.

We already know that cones constitute the right-angled triangle. As per Pythagoras Theorem, length of the slant side, l2 = r2+ h2 Therefore,

Slant height l =√r2+ h2

Furthermore, if a circle whose area is πr2 is used to close the base. Then, the total surface area = πrl +πr2. Therefore,

 Total Surface Area of a Cone = πr (l+r)

Browse more Topics under Surface Areas And Volumes

Volume of Cone

Cones are 3D triangles with a circle-shaped base. According to the structure, the volume of a cone is assumed to be 1/3 of a cylinder with the same radius of base and height. As the volume of a cylinder is πr2h, so,

Volume of a Cone = 1/3 πr2h

where r is the radius of the base and h is the height.

Solved Examples For You

Question: The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28cm, calculate the height, slant height and the curved surface area.

Answer: Volume (V) = 9856 cm3
The diameter of the base = 28 cm.
Radius of the base= d/2 = 28/2= 14 cm
1/3 πr2h = 9856 cm3 = 1/3 × 22/7 × 14 × 14 × h = 9856.
Also, h= (9856 × 3 × 7) /(22 × 14 × 14) = 48 cm
h = 48 cm. l = √r2 + h2 = √ 142 + 482
l = 50 cm.
Curved Surface area = πrl = 22/7 × 14 × 50 = 2200 cm2

Question- How many edges do cones have?

Answer- Zero. Similarly, spheres and cylinders do not have any edges because they do not comprise of flat sides.

Question- What is the difference between a cone and right circular cone?

Answer- All cones may or may not have height to be perpendicular to the radius of the base. However, right circular cones are where the altitude or height is precisely perpendicular to the radius of the base.

Question- Give a few example of cones?

Answer- A funnel, a birthday cap are a few examples of cones.

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