Quiz

1. Find the height of the triangle.

2. Find the length of side c to the nearest meter.

h

c10 m

6 m 9 m

Find the circumference. Use 3.14 for pi.

3. radius 5.6 m

Find the area. Use 3.14 for .

4. diameter 1 ft

Pre-Algebra

6.5

Drawing Three-Dimensional Figures

Find the circumference of each circle, both in terms of and to the nearest tenth.

1. radius 2.5 m

2. diameter 8.8 cm

3. radius 14 ft

Find the area of each circle, both in terms of and to the nearest tenth. Use 3.14 for .

4. diameter 14 ft

196ft2; 615.4 ft2

5m; 15.7 m

8.8cm; 27.6 cm

49ft2; 153.9 ft2

Warm-Up

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

face

edge

vertex

perspective

vanishing point

horizon line

Vocabulary

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.

Edge

Face

Isometric dot paper can be used to draw three-dimensional figures.

Vertex

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.

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.

Example: Continued

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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Step 1: Lightly draw the edges of the bottom face. It will look like a parallelogram. 2 units by 4 units

Try This Continued

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Step 2: Lightly draw the vertical line segments from the vertices of the base. 3 units high

Try This Continued

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Step 3: Lightly draw the top face by connecting the vertical lines to form a parallelogram. 2 units by 4 units

Try This Continued

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Step 4: Darken the lines. Use solid lines for the edges that are visible and dashed lines for the edges that are hidden.

Try This Continued

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.

Sketch a one-point perspective drawing of a cube. V

AB

CD

G

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.

Example: Sketching a One Point Perspective Drawing

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.

Sketch a one-point perspective drawing of a cube.

Example: Sketching a One Point Perspective Drawing

V

A B

C D

G

Step 2: Mark a vanishing point V above your rectangle, and draw a dashed line from each vertex to V.

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.

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.

Try This

A B

C D

G

Step 2: Mark a vanishing point V above your rectangle, and draw a dashed line from each vertex to V.

Step 1: Draw a rectangle. This will be the front face. Label the vertices A through D.

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.

Sketch a one-point perspective drawing of a rectangular box.

Try This

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.

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.

Example: Sketching a Two-Point Perspective Drawing

Darken the visible edges. Erase the horizon and dashed segments.

Sketch a two-point perspective drawing of a cube.

Example: Sketching a Two-Point Perspective Drawing

A

D

V W

C E

Darken the visible edges. Erase the horizon and dashed segments.

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.

Sketch a two-point perspective drawing of a rectangular box.

Label C on DV and E on DW so that CE = ED. Draw vertical segments through C and E. Draw EV and CW.

Try This

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.

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.

Sketch a two-point perspective drawing of a rectangular box.

Try This

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.

Lesson Quiz