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Course: Get ready for AP® Calculus > Unit 4

Lesson 7: Volume and surface area
Math
Get ready for AP® Calculus
Get ready for contextual applications of differentiation
Volume and surface area
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Volume of composite figures

Google Classroom
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

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.

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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
mm3

What is the volume of the cylinder?
  • Your answer should be
  • a multiple of pi, like 12 pi or 2/3 pi
mm3

Subtracting to find the total volume

Let's subtract to find the volume.
What is the remaining volume of the block?
Choose 1 answer:

Adding to find volume

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.

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.

Calculating separate volumes

Let's start with the volume of the triangular prism. All prisms have a volume of (base area)(height).
What is the area of the triangular base?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m2

What is the volume of the triangular prism?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m3

Now we can find the volume of the pyramid. Let’s combine the two half pyramids and find the volume of that whole pyramid.
All pyramids have a volume of 13(base area)(height).
What is the area of the rectangular base of the whole pyramid?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m2

What is the volume of the whole pyramid?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m3

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?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m3

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 190mm long, and the unsharpened base has an area of of 40mm2. The volume of the pencil is 7,040mm3.
We're going to solve for the length p of the sharpened part of the pencil.

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, 7,040mm3 and solve for p.
What is the length p of the sharpened portion of the pencil?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
mm

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