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## High school geometry

### Course: High school geometry>Unit 9

Lesson 3: Volume and surface area

# Volume of composite figures

It's time to mix and match!  Now that we can find volume in several basic shapes, let's combine them to see how we model more interesting and useful real figures.

## Composite volume

A rectangular prism with a hemisphere removed from the top of it.
A prism-like figure that looks like a right rectangular prism, but it has one long edge that is a curved.
A rectangular prism with a triangular prism on the left side of the rectangular prism and a right triangular prism on the right side of the rectangular prism.
It's time to mix and match! Now that we can find volume in several basic shapes, let's combine them to see how we model more interesting and useful real figures.

## Subtracting to find volume

A cylinder has been drilled out of this block to allow a rod to pass through. All measurements are in millimeters. Let's find the remaining volume of the block.
A rectangular prism with a length of seven units, a width of eleven units and a height of nine units. A space in the shape of a cylinder with a radius of three and a length of eleven is in the rectangular prism.

### Subdividing the figure

The figures is composed of a rectangular prism with a cylinder removed from it.

### Finding the separate volumes

What is the volume of the rectangular prism?
start text, m, m, end text, cubed

What is the volume of the cylinder?
start text, m, m, end text, cubed

### Subtracting to find the total volume

Let's subtract to find the volume.
What is the remaining volume of the block?

Let's consider the volume of a tent with the following dimensions. All measurements are in meters. The base of the tent is a rectangle.
A diagram of a tent. The bottom face is a rectangle with a width of twelve units and a length of four units. The front and back faces are quadrilaterals that share an eight unit long edge at the top with each other and that each share a twelve unit long edge with the rectangular face. The top edge of the tent is six units above and parallel to the rectangular face. The left and right faces are triangles that each share a four unit long edge with the rectangular face.

### Subdividing the figure

It looks a little like a triangular prism, but it sticks out too far at the bottom. We can split this figure into the triangular prism and two halves of a pyramid.
The diagram of the tent cut into three figures, one triangular prism and two halves of a pyramid. In the triangular prism, the triangle faces have a base of four units and a height of six units. The height is eight units. In the two pyramid halves, the base length is four units, and the height is six units.

### Calculating separate volumes

Let's start with the volume of the triangular prism. All prisms have a volume of start color #9e034e, left parenthesis, start text, b, a, s, e, space, a, r, e, a, end text, right parenthesis, end color #9e034e, start color #543b78, left parenthesis, start text, h, e, i, g, h, t, end text, right parenthesis, end color #543b78.
What is the area of the triangular base?
start text, m, end text, squared

What is the volume of the triangular prism?
start text, m, end text, cubed

Now we can find the volume of the pyramid. Let’s combine the two half pyramids and find the volume of that whole pyramid.
A pyramid that is the combination of two half pyramids.
All pyramids have a volume of start fraction, 1, divided by, 3, end fraction, start color #9e034e, left parenthesis, start text, b, a, s, e, space, a, r, e, a, end text, right parenthesis, end color #9e034e, start color #543b78, left parenthesis, start text, h, e, i, g, h, t, end text, right parenthesis, end color #543b78.
What is the area of the rectangular base of the whole pyramid?
start text, m, end text, squared

What is the volume of the whole pyramid?
start text, m, end text, cubed

### Adding to find the total volume

Finally, we can add the volumes of the triangular prism and the pyramid to find the total volume of the tent.
What is the volume of the tent?
start text, m, end text, cubed

## Challenge problem

A sharpened pencil is shaped like a right circular cone attached to a cylinder, each with the same radius. The pencil, including the point, is 190, start text, m, m, end text long, and the unsharpened base has an area of of 40, start text, m, m, end text, squared. The volume of the pencil is 7, comma, 040, start text, m, m, end text, cubed.
We're going to solve for the length p of the sharpened part of the pencil.
A pencil modeled by a thin cylinder and a cone attached to one of its bases. The height of the cone has a length of p units. The length of the entire pencil is one hundred ninety units.

### Expressing the volumes in terms of the unknown

Write an expression for the volume of the cone in terms of p.

Write an expression for the height of the cylinder, in terms of p.

Write an expression for the volume of the cylinder, in terms of p.

Write an expression for the entire volume of the pencil, in terms of p.

### Solving for the unknown

Now we can set your expression for the volume of the pencil equal to the numeric volume, 7, comma, 040, start text, m, m, end text, cubed and solve for p.
What is the length p of the sharpened portion of the pencil?
start text, m, m, end text