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

Drawing Three-Dimensional Figures 6.5 Pre-Algebra

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; 615.4 ft2 4. diameter 14 ft 49p ft2; 153.9 ft2

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

Vocabulary face edge vertex perspective vanishing point horizon line

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.

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.

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.

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

Try This Use isometric dot paper to sketch a rectangular box that is 4 units long, 2 units deep, and 3 units tall. • • • • • • • • •

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

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

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

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. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.