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Surface Areas and Volumes

Cylinder

From coffee mugs to gas tanks and fuel containers a cylinder is the most commonly sighted geometric shape. Derived from the Greek word ‘kulindros’ which means a tumbler, a cylinder is the most used shape as well. The chapter below shall deal with the capacity of these most used shapes.

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Cylinder

A cylinder is a closed solid shape with two parallel bases generally circular in shape connected by two parallel sides. A cylinder may also be defined as a rectangle with circular bases. These circular bases are always parallel and congruent to one another. A cylinder when unrolled gives a flattened look of a rectangle. Base and sides together form a cylinder. The line joining the two circular bases at the center is the axis of the cylinder.

 Height (h) as seen in the figure is the perpendicular distance between the two bases. Each base has a radius (r). The measurement of height and radius help us calculate the surface area and volume of the cylinder.

Browse more Topics under Surface Areas And Volumes

Download Cylinder Formula Cheat Sheet Below

Curved Surface area

For calculating the curved surface of this geometrical figure, we need to first wrap a cylindrical can with a sheet of paper. The paper should be wrapped in such a manner with tape that it fits accurately with this cylindrical can.

If needed, clip the paper from top and bottom to match the shape of your Can. Now, remove the paper and cut this cylindrical paper, parallel to its axis, making it rectangular in shape. This activity gives us

Cylinder

[source: tutorvista]

From the figure we know, the length of the cylinder = the circumference of the base = 2πr. The breadth of the rectangle = Height of the cylindrical can = h. Now, since the rectangular paper was used to wrap the Can firmly, we know that area of rectangle = area of the curved surface of the Cylindrical Can = l × b = 2πr × h

The Curved Surface Area of the Cylinder = 2πrh

Cylinder

[source: flickr]

Surface Area

We already know that a cylindrical shape is a combination of two circular bases and rectangle like shape that joins the two bases. So, the total surface area of this geometrical figure shall include the area of the two bases and the curved surface area. Hence, the total surface area = Area of base 1 +Area of base 2+ Curved surface area = πr2 +πr+ 2πrh = 2πr2 + 2πrh = 2πr (r+h). Thus,

The Surface Area of a Cylinder = 2πr (r+h)

Cylinder

[source: socratic]

Volume

A cylindrical shape as said earlier is a built-up of a rectangle with the same size. With the help of two circular bases, we can build a cylinder. Now, for finding the volume of this shape we take the consideration of a 3D counterpart of a rectangle, the cuboid. Using the formula of the volume of a cuboid, we can chart out the volume of any cylindrically shaped container.

We already know that the volume of the cuboid is the product of the area of a rectangle with height; l × b × h. The same is the situation with a 3D cylindrical container. Here we take the product of the base area and height, to get the volume. So, the volume of a cylindrical container = Area of circular base × height = πr× h. Therefore,

The Volume of a Cylinder = πrh

  Also, want to understand Cone?

Solved Examples For You

Question: A restaurant has a large cylindrical vessel of base radius 30 cm filled to a height of 60 cm with milk. The milk is served in small cylindrical glasses of radius 3cm and height 6 cm. Each glass is sold for Rs. 20. How much money does the restaurant earn from serving the milk from this vessel?

Solution: The volume of the milk in the vessel = the total volume of the cylindrical vessel = πR2H

R = 30cm, and H= 60 cm. Here, R is the radius of the cylindrical vessel and H is the height of the vessel from the base = π × 30 × 30 × 60. Likewise, the volume of milk served by each glass = πr2h, r = 3cm and h = 6cm. Here, r and h are the radius and height of cylindrical glass respectively. The volume of each glass = π × 3 × 3 × 6.

As we have to find the total no. of glasses served on the whole day we calculate it using the data as the volume of the cylindrical vessel/ Volume of each glass of milk = π × 30 × 30 × 60 / π × 3 × 3 × 6 = 1000 glasses. Now cost of 1 serving of milk = Rs 20. Cost of 1000 glasses shall be 1000 × 20 = 20,000. So the restaurant earns a total of Rs 20,000 from the mentioned cylindrical vessel.

Ques. What is the formula for a cylinder?

Ans. The formula for calculating the volume of a cylinder is given: V = Bh or V = πr2h. for the result, we have to do the following: the radius of the cylinder we have is 8 cm and the height of the cylinder is given 15 cm. Substitute 8 for r and substitute 15 for h in the formula of V = πr2h.

Ques. Do cylinders have to be round in shape?

Ans. A cylinder is so much similar to a prism. The two bases of the cylinder are circles and not polygons. Moreover, the sides of a cylinder are not flat but curved.

Ques. What are the differences between a circle and a cylinder?

Ans. A 2D geometric figure, a line, comprising of the set of all those points in a plane which are similarly distant from a different point while the cylinder is a surface produced by projecting a closed 2D curve along an axis crossing the plane of the curve.

Ques. Is a cylinder flat or solid?

Ans. A cone is a solid 3D figure that has a round base and a vertex. A cylinder is a solid figure that has 2 parallel congruent circular bases.

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