Three-Dimensional Figures

Vocabulary

• Two-dimensional figures (plane figures) – triangles, quadrilaterals, and circles. They lie in one plane

Vocabulary

• Three-dimensional figures – figures that have length, width, and depth • Cone • Cylinder • Prism • Pyramid • Sphere

• Two-dimensional figures can be transformed through space to create three-dimensional figures. One way is to rotate a two-dimensional figure about a line. This line is sometimes called an axis of symmetry .

Space Figures

• Translate a rectangle through space – It creates a rectangular prism!

Space Figures

• Translate a triangle through space – It makes a triangular prism!

Space Figures

• Translate a circle through space – It makes a cylinder!

Space Figures

• Rotate a rectangle around an axis of symmetry – it also makes a cylinder!

Space Figures

• Twirl (rotate) a triangle around one of its sides – It makes a cone!

Space Figures

• Twirl (rotate) a circle around its diameter – It makes a sphere!

Let’s try some examples!

Example 1

Solution

Example 2

Solution

Example 3

A square measures 1 inch on each side. Suppose a copy of the square is translated 2 inches directly above the plane in which it lies to form a second square. The two squares are then connected with four line segments. Describe the three-dimensional figure that is formed.

Solution

Example 4

Solution

Example 5

Turning Point

• Let’s test your knowledge! Grab your clicker!

1. What is the figure show below called?

1. Cone 2. Cylinder 3. Cube 4. Rectangular prism

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2. The base of a cylinder is shaped like a ______.

1. Rectangle 2. Square 3. Triangle 4. circle

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3

. What three-dimensional figure will be formed if the right triangle shown is rotated about the line shown?

1. Cone

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2. Cylinder 3. Cube 4. sphere

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4. What three-dimensional figure will be formed if the rectangle shown is rotated about the line shown?

1. Cone

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2. Cylinder 3. Cube 4. sphere

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5. A rectangle is translated 3 inches directly above the plane in which it lies. The vertices of the two rectangles are then connected with four line segments. What three dimensional figure is formed?

1. Cone

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2. Cube 3. Cylinder 4. Rectangular prism

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6. What three-dimensional figure will be formed if rectangle ABCD is rotated about AB?

1. Cone

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2. Cube 3. Cylinder 4. prism

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Ticket-out-the-door

• What 3D figure is created when you….

1. Translate a rectangle 2. Translate a circle 3. Rotate a rectangle 4. Translate a triangle 5. Rotate a triangle 6. Rotate a circle 7. Translate a square

Cross-Sections

EQ: How can I describe the cross-sections of solid figures?

Vocabulary

• Cross-section – a view of the inside of a 3D figure after it is sliced • Polyhedron – a 3D figure (a solid with flat faces) http://www.learner.org/channel/courses/learningmath/geometry/se ssion9/part_c/index.html

Example 1

Describe three different cross-sections you can find for the rectangular prism below

Solution

Example 2

The cylinder below sits on a horizontal base. Draw and describe the cross-section formed when the cylinder is cut by a plane that is tilted away from its base.

Solution

• The cross-section is shaped like an oval. In mathematics, this shape is called an ellipse.

Example 3

• The cone below sits on a horizontal base. Draw and describe the cross-sections formed when the cone is cut by a vertical plane through its vertex.

Solution

• The cross-section is shaped like an isosceles triangle. The base of the triangle is on the base of the cone.

Example 4

• The pyramid below has a square horizontal base. Draw and describe the cross-sections formed when the cone is cut by a vertical plane that does not pass through the vertex at its top.

Solution

• The cross-section will be shaped like a quadrilateral. This figure is called an isosceles trapezoid.

Turning Point

• Let’s test your knowledge! Grab your clicker!

1. The cylinder below is cut by the plane shown. What is the shape of the cross section formed?

1. Circle

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2. Rectangle 3. Trapezoid 4. Triangle

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2. The cube below is cut by the plane shown. What is the shape of the cross section formed?

1. Circle

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2. Rectangle 3. Square 4. triangle

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3. Suppose a cone is cut by a plane. Which cross-section is NOT possible?

1. Circle 2. Ellipse 3. Square 4. triangle

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4. The cross-section of a three-dimensional figure is shaped like a circle. The three dimensional figure could NOT be a ________.

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1. Cone 2. Cylinder 3. Pyramid 4. sphere

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5. A cylinder is cut by a plane to form a cross section shaped like an ellipse. How could the plane that formed the cross-section have cut the cylinder?

1. Parallel to a base of the

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cylinder

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2. Perpendicular to a base of the cylinder 3. Slightly tilted away from a base of the cylinder 4. None of the above

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