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In this lesson, we will learn about the volume of a cone.

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Consider a cylinder with the height h, and base radius r.

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Now, we already know that the volume of a cylinder, V, is pi r square h.

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With this, let's change this cylinder into a cone.

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Now, we have a cone with the height h, and base radius r.

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Base on calculations, the volume of the cone found to be one third of a volume of a cylinder.

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Therefore, the volume of a cone, V, is one third pi r square h.

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Let's see some examples on using this formula. For these examples, we take pi as 3.14.

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Find the volume of this cone when its height is 5cm, and its base radius is 3cm.
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Now, we start with the formula for the volume of a cone, V equal 1 third pi r square h.

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Since the radius is 3cm, we can substitute r with 3.

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Now, let's simplify 3 square. 3 square is the same as 3 multiply by 3. This gives 9.

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Let's put this back here.

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Next, since the height is given as 5 cm, we can substitute h with 5.

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We can simplify by multiplying 9 with 5. This gives 45.

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Next, since pi is given as 3.14, we can substitute this pi with 3.14.

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3.14 multiply by 45 gives 141.30.

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Now, this term is the same as, 1 bracket 141.30, over 3.

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1 multiply by 141.30 gives back 141.30.

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141.30 divides by 3, gives 47.10.

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Now, this number has no meaning unless we include the units for it.

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Since the units are given in centimeter, the volume will be in cubic centimeter.

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There, the volume of this cone is 47.10 cubic centimeter.

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Next example, the volume of this cone is 12.56 cubic feet, and its height is 3cm. Find its radius, r.

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To solve this, we start with the formula for the volume of a cone, V equals 1 third pi r square h.

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Here, we can see that since the volume, pi, and h are given, we can find r, by solving this equation for r. Here’s how.

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First, it is easier to work with this equation if we write 1 third pi r square h in the form of fraction. By doing so, we have, 1 pi r square h, over 3.

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1 pi r square h is the same as, pi r square h.

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Next, note that we can remove the fraction in this equation, by multiplying both sides of the equation with 3.

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This gives, 3 'V' equals to pi r square h.

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Now, the volume of the cone is 12.56 ft. Hence, we can substitute V with 12.56.

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3 multiply by 12.56 gives 37.68

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Next, since the height of the cone is 3 ft, we can substitute h with 3.

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Similarly, since pi is given as 3.14, we can substitute this pi with 3.14.

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We can simplify by multiplying 3.14 with 3. This gives 9.42.

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Alright, now we have 9.42 r square, equals to, 37.68.

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Let's rewrite this, so it will look neater.

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To get closer to finding r, we need to remove 9.42.

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We can do so, by dividing both sides of the equation with 9.42. This gives, r square equals to 37.68 over 9.42.

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37.68 divides by 9.42, gives 4.

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Now, since r square is equals to 4, we can find r, by calculating the square root of 4.

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Square root of 4 gives 2.

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Again, this number has no meaning unless we include the unit for it.

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Since the volume is in cubic feet, the radius of the base of this cone will be in feet.

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Hence, the radius of the base of this cone is 2 ft.

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This is all for this lesson. Try out the practice question to further your understanding.