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Chapter 11: Surface Areas and Volumes

    Surface Area of a Right Circular Cone

    Surface Area of a Sphere

    Volume of a Right Circular Cone

    Volume of a Sphere

    Exercise 11.1

    Exercise 11.2

    Exercise 11.3

    Exercise 11.4

    Summary

    Enhanced Curriculum Support

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Chapter 11: 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 = 13 πr2h

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

Example 8: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

Instructions

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.

Example 9: 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.

Instructions

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.