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Surface Areas and Volumes

    Volume of a Right Circular Cone

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

Volume of a Right Circular Cone

Having covered the surface area, let's move to volume of the discussed shapes.

Activity: Try making a hollow cone and cylinder having the same base radius and height. Fill the cone with sand upto the brim and pour it into the cylinder. The cylinder will be filled up partially. Repeat this step until the whole cylinder is filled up to the top.

As it so happens, we need to make the transfer of sand from the cone to the cylinder exactly three times. When we compare the volumes of a right cylinder and a right circular cone (having the same radius and height), we reach the conclusion that the volume of the cylinder is three times that of the cone. In other words, the volume of the cone is one-third the volume of the cylinder. Since, we already know that:

Volume of cylinder = πr2h we get

Volume of a Cone = \frac{1}{3} πr2h

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

Let's Solve

  1. The height and the slant height of a cone are 21 cm and 28 cm respectively. The volume of the cone is _____

Note: l2 = r2+h2

l2=r2+h2

  • Let's start by finding the radius by substituting values
  • The radius of the cone:
  • We know volume of cone = where r and h are the radius and height of the cone.
  • Substituting the values of r and h into the formula, we get: cm3
  • We have found the desired answer.

  1. Monica has a piece of canvas whose area is 551 m2. She uses it to have a conical tent made, with a base radius of 7 m. Assuming that all the stitching margins and the wastage incurred while cutting amounts to approximately 1 m2, find the volume of the tent that can be made with it.

Finding volume of tent

  • The area of the canvas = m2 and area of the canvas lost in wastage = m2. Thus, the area of canvas available for making the tent = m2
  • Now, we need to find the of the tent.
  • We know curved surface of tent = where r and l are radius and slant height.
  • Finding the value of slant height, we get: l = m
  • Using l2 = r2+h2, we can find the height by substituting values
  • Height h = m.
  • We know volume of cone =
  • Finding the volume, we get: m3
  • We have found the desired answer.

  1. A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?

Finding volume of pit

  • From the given data, we get: Radius of pit (r) = m
  • We know volume of pit = where r and h are the radius and height,respectively.
  • Substituting values
  • We get volume of pit = m3
  • We know 1 m3 = L = kL
  • Thus, the capacity of the pit is kiloliters (kL).
  • We have found the desired answer.

  1. A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.

Finding volume of solid

  • Upon revolving the right triangle, we get
  • We know volume of solid = where r and h are the radius and height,respectively.
  • Substituting values
  • We get volume of solid = × π cm3
  • We have found the desired answer.

  1. If the triangle ABC in the previous question, was revolved about the side 5 cm, then find the volume of the solid so obtained. Also find the ratio of the volumes of the two solids obtained in the previous and current case.

Finding volume of solid

  • In this case, we obtain a cone with height = cm and radius = cm which further gives the slant height = cm.
  • Substituting values in the volume formula
  • We get the volume = π cm3
  • Finding the ratio of PreviousCaseCurrentCase = or =
  • We have found the desired answer.