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Surface Areas and Volumes

    Surface Area of a Right Circular Cone

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9th class > Surface Areas and Volumes > Surface Area of a Right Circular Cone

Surface Area of a Right Circular Cone

In the earlier grades, having studied the surface areas of basic shapes like cube, cuboid and cylinder, let's move on to study the surface area of cone.

We have upto now, been generating solids by stacking up congruent figures. A prisms is one such example. Let's look at another kind of solid which is not a prism. These kinds of solids are called pyramids.

Activity: Cut out a right-angled triangle from a piece of paper. Paste the cutout on a long thick string along one of the perpendicular sides of the triangle i.e. the side that corresponds as the height. Now, hold the string with hands on either sides of the cutout and rotate the triangle using the string a number of times.

The shape that is generated is a right circular cone. A right angled triangle, when rotated to a full 360o degrees gives a a right circular cone.

The'h' and 'r' denote the height and radius of the cone while 'l' is called the slant height of the cone i.e. the length of the curved surface's slope.

Since, the cone is right angled, we also get that:

l2 = r2 + h2

We know that the base of the cone is a circle. Thus, the area of the base can be evaluated. Now, how can the area of the entire curved surface be found?

Activity: Take a cone which doesn't have any operlapping surface. If unavailable, make a cone from a piece of paper. Cut out the cone along the slant-height .Upon opening it out and flattening the material, we will see a circular shape with a sector cut out of it (resembling a piece of cake which has a portion of it cutout).

Now, mark and cutout this circular shape into smaller triangles, all having their height cut out along the original slant height of the cone. Which in this case,is the radius of the circular shape. Now, individually, measure and calculate the area of each triangle.

Area of each triangle = \frac{1}{2} × base of each triangle × slant height

Thus we can say,

Area of the entire curved surface = sum of all the smaller triangles

= \frac{1}{2} x slant height x (b1 + b2 + .....)

= \frac{1}{2} x slant height x (circumference of the partial circle)

= \frac{1}{2} x slant height x (2πr)

= πr x slant height = πrl

Note: The curved portion of the figure makes up the perimeter of the base of the cone and the circumference of the base of the cone becomes 2πr, where r is the base radius of the cone.

Now if the base of the cone is to be closed, then a circular piece of paper of radius r is also required whose area is πr2. Thus,

So we have,

Curved Surface Area of a Cone = πrl

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

Total Surface Area of a Cone = πrl + πr2 = πr(l + r)

Let's Solve

  1. The height of a cone is 16 cm and its base radius is 12 cm. Find the curved surface area and the total surface area of the cone (Use π = 3.14)

Note: Round off the decimals to the nearest whole number

Finding surface area

  • Since, l2= h2+r2 we find the value of slant height l.
  • Calculating, we get l = cm.
  • We know Curved surface area =
  • Calculating, we get the value to be cm2(Round off to the nearest whole number)
  • We also know, total surface area = where l and r have the usual dimensional abbreviations.
  • Substitute the values in the eq., we get the answer as cm2(Round off to the nearest whole number)
  • This gives us the value
  • We have found the desired answers.

  1. A corn cob, shaped somewhat like a cone, has the radius of its broadest end as 2.1 cm and length (height) as 20 cm. If each 1 cm2 of the surface of the cob carries an average of four grains, how many grains are there on the entire cob.

Finding number of grains

  • The corn grains cover the of the corn cob.
  • We know that is equal to where l - slant height and r - radius.
  • Calculating the value of l
  • We find the value of l = cm (Upto two decimal places)
  • Calculating the numerical value of the surface area
  • We find the value to be cm2 (Upto two decimal places)
  • We have been given that the number of grains of corn on 1 cm2 of the surface of the corn cob is 4.
  • Thus, number of grains on the cob = (Enter the nearest whole number)
  • So, approximately 531 grains of corn are present on the cob.

  1. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m?

Note: Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use π = 3.14)

Finding length of material

  • Using given values to calculate value of slant height
  • We find the value to be m.
  • We know that the curved surface of tent is equal to where l - slant height and r - radius.
  • Substituting the values
  • We find curved surface of tent = m2(Round off to one decimal place)
  • We have been given that 20 cm is wastage provision for the cloth.Let the length of the tarpaulin sheet required is L. Then, the effective length used for the tent is m
  • We have been given: breadth of tarpaulin = 3m. We also know: Area of material sheet = of the tent
  • Putting the values
  • Upon equating we get L = m
  • Thus, length of the required tarpaulin sheet will be 63 m.

  1. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of Rs 210 per 100 m2

Finding cost of whitewashing

  • The base radius (r) of the tent is = m
  • We know that the curved surface of tent is equal to where l - slant height and r - radius.
  • Substituting the values, we get: m2
  • Using the cost of the wahitewashing per 100m2, we can calculate the total cost.
  • We find the total cost to be Rs.
  • Thus, we have found the desired answer.