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The volume of a solid is a measure of how much space it occupies in three dimensions. Meaning, it is the three-dimensional equivalent of area. Volume is measured using cubic units, such as cubic meters, m$_{3}.$

Cavalieri's principle states that two solids with the same height and the same cross-sectional area at every height have the same volume. This means, for instance, that skewed versions of the same solid have the same volume, so long as their heights are equal.

The volume of a solid can be calculated with a formula that depends on the solid's shape.

The volume of a prism is given by multiplying the area of its base, $B,$ with its height, $h.$ $V=Bh$

The volume of a pyramid is a third of the volume of a prism with the same base. Thus, the formula for the pyramid's volume is: $V=3Bh .$

The formula for the volume of a cylinder can be proven by placing a rectangular prism next to it.

Suppose the prism and the cylinder have the same base area and height.

Thus, Cavalieri's principle states that two solids with the same base area and height have the same volume. Since the volume of a prism is given by $V_{P}=Bh,$ the volume of a cylinder can be calculated with the same formula. The base of a cylinder has the shape of a circle. Its area can be expressed as $B=πr_{2}.$ Therefore, the formula for the volume of a cylinder is $V_{C}=πr_{2}h.$

It applies to all cylinders because there is always a prism with the same base area and height.The formula for the volume of a cone can be derived using a pyramid.

Assume, the pyramid and cone have the same height, $h,$ and that the pyramid has a square base, whose side length is twice the radius of the cone's base.

Since the cone's diameter is equal to the side length of the square, the cone will fit inside the pyramid.

The base of the cone is a circle that fits exactly in the base of the pyramid. By studying the figure from below, the ratio between the bases can be determined.

The area of the square is calculated by taking the square of the side length. Further, the area of the circle is given by the square of the radius times $π.$
$A_{S}=(2r)_{2}A_{C}=πr_{2}$
The ratio between the areas can now be calculated by dividing the area of the circle with the area of the square.

$A_{S}A_{C} $

SubstituteExpressionsSubstitute expressions

$(2r)_{2}πr_{2} $

CalcPowCalculate power

$4r_{2}πr_{2} $

SimpQuotSimplify quotient

$4π $

$V_{C}=4π ⋅3(2r)_{2}h $

CalcPowCalculate power

$V_{C}=4π ⋅34r_{2}h $

MultFracMultiply fractions

$V_{C}=4⋅34⋅πr_{2}h $

SimpQuotSimplify quotient

$V_{C}=3πr_{2}h $

A sphere is a perfectly round solid composed of many circles. The formula for the volume of the sphere is given by: $V=34πr_{3} ,$

where $r$ is the radius of the largest circle in the sphere.A solid that is made up of more than one solid is called a composite solid. The individual solids can be combined either by adding or subtracting them from one another. For instance, a hemisphere — half a sphere — can be combined with a cone to make something that resembles a snow cone, or it could be used to hollow out a cylinder.

The volume of a composite solid is either the sum or difference between the volumes of the individual solids, whichever is applicable.

Find the volume of ice cream in the wafer cup.

Assume that the wafer is in the shape of a cone and that the ice cream above the wafer is a hemisphere.

Show Solution

The ice cream can be interpreted as a composite solid, a combination of a cone and a hemisphere.

Thus, the sum of the cone's volume and the hemisphere's volume will be the volume of the ice cream. Let's start with calculating the volume of the cone.

The volume of a cone can be calculated with the formula $V=3Bh ,$ where $B$ is the area of the base and $h$ is the height. The base of a cone is in the shape of a circle, Its area can be calculated using $B=πr_{2}.$ The radius of the base is the same as for the hemisphere, since it fits exactly with the cone.

The height of the cone is $4$ inch and the radius is $1$ inch. By substituting the known values, the volume of the cone can be calculated.
The volume of the cone is $34π in_{3}$.

A hemisphere is half of a sphere. Therefore, to calculate the volume we can use the formula for the volume of a sphere and multiply it by $21 .$ $V=34πr_{3} ⋅21 =64πr_{3} $ The formula requires the radius of the hemisphere, which is $1$ inch.

Now, the volume of the hemisphere can be calculated.
Thus, the volume of the hemisphere is $32π in_{3}.$

$V=V_{C}+V_{H}$

$V=34π +32π $

$V=2π$

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