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In this lesson, we will look at several different three-dimensional shapes. You will learn how to find the volumes of these shapes.
A green rectangle is drawn to the left. It is 2 units wide and 4 units long.
Because each slice is 1 unit tall, the number of unit cubes in each slice is the same as the number of unit squares in the base (the area of the base). So the volume of the entire shape is equal to the area of its base, multiplied by its height.
A shape that can be built in this way — by taking a base which is a polygon and extending it straight up — is called a right prism. A right prism with a triangular base is drawn to the left, along with its base. You can click and drag up and down to rotate the prism, to see its top and bottom better.
As you saw in prism-qn, you can find the volume of a right prism by multiplying the area of its base by its height:
If a right prism’s base has area $b$ and the prism has height $h$, then its volume is $bh$.
A right prism whose base is a star is drawn to the left. The base alone is drawn below the prism.
In fact, the area of the base of the prism is almost exactly 18 square units.
The base of a right prism has to be a polygon. A cylinder is like a right prism, except that its base is a circular disk. There is a cylinder drawn to the left, along with its base. The radius of the base (distance from its center to its edge) is 4.
The volume of a cylinder can be found by multiplying the area of its base by its height. Since a circle with radius $r$ has an area of $πr^2$, this means that:
If the base of a cylinder has radius $r$, and the cylinder has height $h$, its volume is $πr^2h$.
A pyramid is the shape formed by taking a base which is a polygon and extending it up to a point. A pyramid with a square base is drawn to the left. Each side of the square base is 8 units long, and the pyramid is 4 units tall.
You can slide the slider to split the pyramid into four unit slices. What is the approximate volume of each of these slices? (Estimate the number of unit cubes each one contains, guessing very roughly at the volume of partial unit cubes.)
The blue pyramid to the left has a square base with sides that are $s$ units long. The pyramid is $$s/2$$ units tall. You can click and drag left and right to rotate it. We would like to find its volume.
Click to see a multicolored cube. You can slide the slider to unfold this cube into 6 copies of the blue pyramid.
Notice that this is the same as the volume of the pyramid that you found above. So you have seen that this pyramid has a volume of $$1/3 b h$$, where $b$ is the area of its base and $h$ is its height. In fact this is true of any pyramid:
If the base of a pyramid has area $b$, and the pyramid has height $h$, its volume is $1/3 b h$.
A cone is like a pyramid, except that its base is a circular disk. There is a cone drawn to the left, along with its base. The base of the cone has a radius of 3, and the cone has a height of 5.
If the base of a cone is a circular disk with radius $r$, the base has an area of $πr^2$. This means that:
If the base of a cone has radius $r$, and the cone has height $h$, its volume is $$1/3 π r^2 h$$.
A sphere consists of all the points in three dimensions which are at the same distance (the sphere’s radius) from a central point. A sphere with a radius of 3 is drawn to the left.
You can slide the slider to split the sphere into six unit slices. What is the approximate volume of each of these slices?
What is the approximate total volume of the sphere?
A sphere with radius $r$ is drawn to the left. You can click to show, and then hide, a cylinder with base radius $r$ and height $2r$ around the sphere.
The cylinder has a base radius of $r$ and a height of $2r$, so it has a volume of $πr^2(2r)=2πr^3$. The cones each have a base radius of $r$ and a height of $r$, so they each have a volume of $1/3πr^2(r)$. So their total volume is $1/3πr^2(r)+1/3πr^2(r)=2/3πr^3$.
This means that the volume of the sphere must be somewhere in between $2/3πr^3$ and $2πr^3$. It turns out to be exactly halfway between these two numbers:
The volume of a sphere with radius $r$ is $4/3πr^3$.