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Composite Solids

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What if you built a solid three-dimensional house model consisting of a pyramid on top of a square prism? How could you determine how much two-dimensional and three-dimensional space that model occupies? After completing this Concept, you'll be able to find the surface area and volume of composite solids like this one.

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Composite Solids CK-12

Guidance

A composite solid is a solid that is composed, or made up of, two or more solids. The solids that it is made up of are generally prisms, pyramids, cones, cylinders, and spheres. In order to find the surface area and volume of a composite solid, you need to know how to find the surface area and volume of prisms, pyramids, cones, cylinders, and spheres. For more information on any of those specific solids, consult the concept that focuses on them. This concept will assume knowledge of those five solids.

Most composite solids problems that you will see will be about volume, so most of the examples and practice problems below are about volume. There is one surface area example as well.

Example A

Find the volume of the solid below.

This solid is a parallelogram-based prism with a cylinder cut out of the middle.

V_{prism} &= (25 \cdot 25)30=18,750 \ cm^3\\V_{cylinder} &= \pi (4)^2 (30)=480 \pi \ cm^3

The total volume is 18750 - 480 \pi \approx 17,242.04 \ cm^3 .

Example B

Find the volume of the composite solid. All bases are squares.

This is a square prism with a square pyramid on top. First, we need the height of the pyramid portion. Using the Pythagorean Theorem, we have, h=\sqrt{25^2-24^2}=7 .

V_{prism} &= (48)(48)(18)=41,472 \ cm^3\\V_{pyramid} &= \frac{1}{3} (48^2)(7)=5376 \ cm^3

The total volume is 41,472 + 5376 = 46,848 \ cm^3 .

Example C

Find the surface area of the following solid.

This solid is a cylinder with a hemisphere on top. It is one solid, so do not include the bottom of the hemisphere or the top of the cylinder.

SA &=LA_{cylinder}+LA_{hemisphere}+A_{base \ circle}\\&= 2 \pi rh+\frac{1}{2} 4 \pi r^2+\pi r^2\\&= 2 \pi (6)(13)+2 \pi 6^2+\pi 6^2\\&= 156 \pi +72 \pi +36 \pi\\&= 264 \pi \ in^2 \qquad \qquad ``LA'' \ \text{stands for} \ lateral \ area.

Composite Solids CK-12

Guided Practice

1. Find the volume of the following solid.

2. Find the volume of the base prism. Round your answer to the nearest hundredth.

3. Using your work from #2, find the volume of the pyramid and then of the entire solid.

Answers:

1. Use what you know about cylinders and spheres. The top of the solid is a hemisphere.

V_{cylinder} &= \pi 6^2 (13)=468 \pi\\V_{hemisphere} &= \frac{1}{2} \left(\frac{4}{3} \pi 6^3\right)=144 \pi\\V_{total} &= 468 \pi+144 \pi =612 \pi \ in^3

2. Use what you know about prisms.

V_{prism}&=B \cdot h \\  V_{prism}&=(4\cdot 4)\cdot 5\\ V_{prism}&=80in^3

3. Use what you know about pyramids.

V_{pyramid}&=\frac{1}{3} B \cdot h \\ V_{pyramid}&=\frac{1}{3}(4 \cdot 4)(6)\\ V_{pyramid}&=32in^3

Now find the total volume by finding the sum of the volumes of each solid.

V_{total}&=V_{prism}+V_{pyramid}\\ V_{total}&=112 in^3

Practice

Round your answers to the nearest hundredth. The solid below is a cube with a cone cut out.

  1. Find the volume of the cube.
  2. Find the volume of the cone.
  3. Find the volume of the entire solid.

The solid below is a cylinder with a cone on top.

  1. Find the volume of the cylinder.
  2. Find the volume of the cone.
  3. Find the volume of the entire solid.
  1. You may assume the bottom is open .

Find the volume of the following shapes. Round your answers to the nearest hundredth.

  1. A sphere has a radius of 5 cm. A right cylinder has the same radius and volume. Find the height of the cylinder.

The bases of the prism are squares and a cylinder is cut out of the center.

  1. Find the volume of the prism.
  2. Find the volume of the cylinder in the center.
  3. Find the volume of the figure.

This is a prism with half a cylinder on the top.

  1. Find the volume of the prism.
  2. Find the volume of the half-cylinder.
  3. Find the volume of the entire figure.

Tennis balls with a 3 inch diameter are sold in cans of three. The can is a cylinder. Round your answers to the nearest hundredth.

  1. What is the volume of one tennis ball?
  2. What is the volume of the cylinder?
  3. Assume the balls touch the can on the sides, top and bottom. What is the volume of the space not occupied by the tennis balls?

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