# Solid Geometry

In this lesson, we will look at several different three-dimensional shapes. You will learn how to find the volumes of these shapes.

## Right prisms and cylinders

A green rectangle is drawn to the left. It is 2 units wide and 4 units long.

 What is the area of the rectangle? (That is, what is the total number of green unit squares required to fill it?)
 Click . Now the rectangle has been extended upward to form a rectangular box which is 1 unit tall. What is the volume of this box? (That is, what is the total number of green unit cubes required to fill it?)
 Is the volume of the 1-unit-tall box (in unit cubes) equal to the area of the rectangle (in unit squares)?
 Click . Now the rectangular box is 3 units tall. You can slide the slider to slice the box into shorter boxes which are 1 unit tall. What is the volume of the entire box (that is, all of its slices put together)?

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 can see, the base of this prism is a right triangle whose legs have lengths 3 and 6. What is its area? (The area of a right triangle whose legs have lengths \$a\$ and \$b\$ is \$1/2 ab\$.)
 Slide the slider to split the prism into unit slices (slices of height 1). What is the volume of a single unit slice of the prism?
 What is the height of this prism? That is, how many unit triangular slices make up the prism?

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\$.

 What is the volume of the entire prism?

A right prism whose base is a star is drawn to the left. The base alone is drawn below the prism.

 By counting the number of green squares in the base of this prism, estimate its area.
 What is the height of this prism?

In fact, the area of the base of the prism is almost exactly 18 square units.

 Using this information, what is the approximate volume of a single unit slice of the prism?
 What is the approximate volume of the entire prism?

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.

 What is the area of the base? Round your final answer to two decimal places. (The area of a circle with radius \$r\$ is \$πr^2\$, where \$π\$ is a number that is approximately 3.1416.)
 What is the height of this cylinder?
 What is the volume of the cylinder? Round your answer to one decimal place.

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 cylinder which is 6.4 units tall and has a base radius of 2.6 units is drawn to the left. What is the volume of that cylinder? Round your answer to one decimal place.
 You have a cylindrical barrel which is 6.4 feet tall and has a base radius of 2.6 feet. How much water will the barrel hold? (That is, what is its volume?) Round your answer to one decimal place. cubic feet

## Pyramids and cones

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.)

SliceVolume
 What is the approximate total volume of the pyramid?

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.

 The volume of the cube is \$s^3\$. What is the volume of each of the 6 pyramids? (Because the 6 pyramids are all the same size and shape, they must split the volume of the cube up into equal parts.)
 The green pyramid from pyramid-slice-qn has the same shape as the blue pyramid, with \$s=8\$. What is its volume, rounded to one decimal place?
 Is this close to the approximation of the volume you found in pyramid-slice-qn?
 The area of the base of the pyramid is \$s^2\$. The height of the pyramid is \$\$s/2\$\$. What do you get if you multiply \$\$1/3\$\$ of the base area times the height?

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 right triangle is drawn to the left. Its legs have lengths 3 and 4. What is its area?
 Click to see a pyramid whose base is that right triangle. You can click and drag up and down to rotate the pyramid, to see its base better. What is the height of this pyramid? (That is, how many slices does the slider split it into?)
 Using the volume formula from pyramid-decomp-qn, find the volume of this pyramid.

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.

 What is the area of the base? Round your answer to two decimal places.
 A cone is very similar to a pyramid. As with a pyramid, if a cone’s base has area \$b\$ and the cone has height \$h\$, its volume is \$\$1/3 b h\$\$. What is the volume of this cone? Round your answer to one decimal place.

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\$\$.

 Click to see a cone with a base radius of 4 and a height of 2. What is its volume? Round your answer to one decimal place.

## Spheres

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?

SliceVolume

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.

 Does the cylinder completely surround the sphere?
 Which shape has a larger volume: the cylinder or the sphere?
 Click to see two cones with base radius \$r\$ and height \$r\$. You can click to compare the cones with a sphere of radius \$r\$. Does the sphere completely surround the two cones?
 Which shape has a larger volume: the two cones or 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\$.

 What is the volume of a sphere with radius 3? Round your answer to one decimal place.
 Is this close to the approximation of the volume you found in sphere-slice-qn?
 A sphere has a radius of 6. What is its volume? Round your answer to one decimal place.
 Isaac blows a spherical bubble with a radius of 6 centimeters. How much air did he blow into the bubble? (That is, what is its volume?) Round your answer to one decimal place. cubic centimeters