1. Quiz 10 m c 1. Find the height of the triangle. h
2. Find the length of side c to the nearest meter.
6 m
9 m
Find the circumference. Use 3.14 for pi.
3. radius 5.6 m
Find the area. Use 3.14 for p.
4. diameter 1 ft

2. Drawing Three-Dimensional Figures
6.5
Pre-Algebra

3. Warm-Up
Find the circumference of each circle, both in terms of p and to the nearest tenth.
1. radius 2.5 m
5p m; 15.7 m
2. diameter 8.8 cm
8.8p cm; 27.6 cm
Find the area of each circle, both in terms of p and to the nearest tenth. Use 3.14 for p.
3. radius 14 ft
196p ft2; ft2
4. diameter 14 ft
49p ft2; ft2

4. Learn to draw and identify the parts of three-dimensional figures.

5. Vocabulary
face
edge
vertex
perspective
vanishing point
horizon line

6. Three-dimensional figures have faces, edges, and vertices
Three-dimensional figures have faces, edges, and vertices. A face is a flat surface, an edge is where two faces meet, and a vertex is where three or more edges meet.

7. Isometric dot paper can be used to draw three-dimensional figures.
Vertex
Face
Edge
Isometric dot paper can be used to draw three-dimensional figures.

8. Example: Drawing a Rectangular Box
Use isometric dot paper to sketch a rectangular box that is 5 units long, 3 units deep and 2 units tall.
Step 1: Lightly draw the edges of the bottom face. It will look like a parallelogram. 3 units by 5 units
Step 2: Lightly draw the vertical line segments from the vertices of the base. 2 units high
Step 3: Lightly draw the top face by connecting the vertical lines to form a parallelogram. 3 units by 5 units
Step 4: Darken the lines. Use solid lines for the edges that are visible and dashed lines for the edges that are hidden.

9. Example: Continued
Use isometric dot paper to sketch a rectangular box that is 5 units long, 3 units deep and 2 units tall.

10. Try This
Use isometric dot paper to sketch a rectangular box that is 4 units long, 2 units deep, and 3 units tall.
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11. Try This Continued
Step 1: Lightly draw the edges of the bottom face. It will look like a parallelogram. 2 units by 4 units
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12. Try This Continued
Step 2: Lightly draw the vertical line segments from the vertices of the base. 3 units high
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13. Try This Continued
Step 3: Lightly draw the top face by connecting the vertical lines to form a parallelogram.
2 units by 4 units
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14. Try This Continued
Step 4: Darken the lines. Use solid lines for the edges that are visible and dashed lines for the edges that are hidden.
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15. Perspective is a technique used to make drawings of three-dimensional objects appear to have depth and distance. In one-point perspective drawings, there is one vanishing point.

16. Example: Sketching a One Point Perspective Drawing
Sketch a one-point perspective drawing of a cube. V
Step 1: Draw a square. This will be the front face. Label the vertices A through D. G A B C D
Step 2: Mark a vanishing point V above your square, and draw a dashed line from each vertex to V.
Step 3: Choose a point G on BV and draw a smaller square that has G as one of its vertices.
Step 4: Darken the visible edges, and draw dashed segments for the hidden edges. Erase the vanishing point and the lines connecting it to the vertices.

17. Example: Sketching a One Point Perspective Drawing
Sketch a one-point perspective drawing of a cube.
Step 1: Draw a square. This will be the front face. Label the vertices A through D.
Step 2: Mark a vanishing point V above your square, and draw a dashed line from each vertex to V.
Step 3: Choose a point G on BV and draw a smaller square that has G as one of its vertices.
Step 4: Darken the visible edges, and draw dashed segments for the hidden edges. Erase the vanishing point and the lines connecting it to the vertices.

18. Sketch a one-point perspective drawing of a rectangular box.
Try This
Sketch a one-point perspective drawing of a rectangular box. V
Step 1: Draw a rectangle. This will be the front face. Label the vertices A through D. G A B C D
Step 2: Mark a vanishing point V above your rectangle, and draw a dashed line from each vertex to V.
Step 3: Choose a point G on BV and draw a smaller rectangle that has G as one of its vertices.
Step 4: Darken the visible edges, and draw dashed segments for the hidden edges. Erase the vanishing point and the lines connecting it to the vertices.

19. Sketch a one-point perspective drawing of a rectangular box.
Try This
Sketch a one-point perspective drawing of a rectangular box.
Step 1: Draw a rectangle. This will be the front face. Label the vertices A through D. G A B C D
Step 2: Mark a vanishing point V above your rectangle, and draw a dashed line from each vertex to V.
Step 3: Choose a point G on BV and draw a smaller rectangle that has G as one of its vertices.
Step 4: Darken the visible edges, and draw dashed segments for the hidden edges. Erase the vanishing point and the lines connecting it to the vertices.

20. You can also draw a figure in two-point perspective by using two vanishing points and a horizon line.
Moving the horizon line up and down gives you different views of the figure.

21. Example: Sketching a Two-Point Perspective Drawing
Sketch a two-point perspective drawing of a cube.
Draw a vertical segment AD. Draw a horizontal line above AD and label the vanishing points V and W. Draw dashed segments AV, AW, DV, and DW.
Label C on DV and E on DW so that CE = ED. Draw vertical segments through C and E. Draw EV and CW.

22. Example: Sketching a Two-Point Perspective Drawing
Sketch a two-point perspective drawing of a cube.
Darken the visible edges. Erase the horizon and dashed segments.

23. Sketch a two-point perspective drawing of a rectangular box.
Try This
Sketch a two-point perspective drawing of a rectangular box. V W C E A D
Draw a vertical segment AD. Draw a horizontal line above AD and label the vanishing points V and W. Draw dashed segments AV, AW, DV, and DW.
Label C on DV and E on DW so that CE = ED. Draw vertical segments through C and E. Draw EV and CW.
Darken the visible edges. Erase the horizon and dashed segments.

24. Sketch a two-point perspective drawing of a rectangular box.
Try This
Sketch a two-point perspective drawing of a rectangular box.
Draw a vertical segment AD. Draw a horizontal line above AD and label the vanishing points V and W. Draw dashed segments AV, AW, DV, and DW.
Label C on DV and E on DW. Draw vertical segments through C and E. Draw EV and CW.
Darken the visible edges. Erase the horizon and dashed segments.

25. Lesson Quiz
1. Use isometric dot paper to sketch a rectangle box 3 units tall with a base of 2 units by 5 units.
2. Sketch a cube in one-point perspective.
3. Sketch a brick in two-point perspective.