Unit 9 Understanding 3D Figures Geometry 2014-15.

Unit 9 Understanding 3D Figures Geometry 2014-15

Unit Outline 10.1 Areas of Parallelograms and Triangles (0.5 day) 10.2 Areas of Trapezoids, Rhombuses, and Kites (1.5 day) 10.3 Areas of Regular Polygons (2 days) 11.1 Space Figures and Cross Sections (2 days) 11.4 Volumes of Prisms and Cylinders (1 day) 11.5 Volumes of Pyramids and Cones (2 days) 11.6 Surface Area and Volumes of Spheres (1 day) 11.7 Areas and Volumes of Similar Solids (2 days) 10.1 Areas of Parallelograms and Triangles (0.5 day) 10.2 Areas of Trapezoids, Rhombuses, and Kites (1.5 day) 10.3 Areas of Regular Polygons (2 days) 11.1 Space Figures and Cross Sections (2 days) 11.4 Volumes of Prisms and Cylinders (1 day) 11.5 Volumes of Pyramids and Cones (2 days) 11.6 Surface Area and Volumes of Spheres (1 day) 11.7 Areas and Volumes of Similar Solids (2 days)

10.1 Areas of Parallelograms and Triangles Unit 9 Understanding 3D Figures

Vocabulary

Finding the Area of a Parallelogram

Find the Missing Measurement

Your Turn! Find the area of each parallelogram. Find the value of h for each parallelogram.

Vocabulary

Finding the Area of a Triangle A triangle has an area of 18 in 2. The length of its base is 6 in. What is the corresponding height? Solution: Draw a sketch of the triangle to visualize the problem. A = bh Substitute 18 = (6)hSimplify 18 = 3h h = 6 in The height of the triangle is 6 in. A triangle has an area of 18 in 2. The length of its base is 6 in. What is the corresponding height? Solution: Draw a sketch of the triangle to visualize the problem. A = bh Substitute 18 = (6)hSimplify 18 = 3h h = 6 in The height of the triangle is 6 in.

Your Turn! A triangle has height 11 in. and base length 10 in. Find its area. A triangle has area 24 m 2 and base length 8 m. Find its height. The figure at the right consists of a parallelogram and a triangle. What is the area of the figure? A triangle has height 11 in. and base length 10 in. Find its area. A triangle has area 24 m 2 and base length 8 m. Find its height. The figure at the right consists of a parallelogram and a triangle. What is the area of the figure?

Questions? Instructor Email: [email protected]@lake.k12.fl.usInstructor Email: [email protected]@lake.k12.fl.us

10.2 Areas of Trapezoids, Rhombuses, and Kites Unit 9 Understanding 3D Figures

Vocabulary

Finding the Area of a Trapezoid

Your Turn! Find the area of each trapezoid. If necessary, leave your answer in simplest radical form.

Vocabulary

Finding the Area of a Kite

Finding the Area of a Rhombus

Your Turn! Find the area of each figure. Leave your answer in simplest radical form.

10.3 Areas of Regular Polygons Unit 9 Understanding 3D Figures

Vocabulary Radius of a Regular Polygon Distance from the center to a vertex Apothem Perpendicular distance from the center to a side Radius of a Regular Polygon Distance from the center to a vertex Apothem Perpendicular distance from the center to a side

Finding Angle Measures

Your Turn! Each regular polygon has radii and apothem as shown. Find the measure of each numbered angle.

Vocabulary

Finding the Area of a Regular Polygon

Your Turn! Find the area of each regular polygon with the given apothem, a, and side length, s. pentagon, a = 4.1 m, s = 6 m octagon, a = 11.1 ft, s = 9.2 ft Find the area of each regular polygon with the given apothem, a, and side length, s. pentagon, a = 4.1 m, s = 6 m octagon, a = 11.1 ft, s = 9.2 ft

Your Turn! Find the area of each regular polygon. Round your answer to the nearest tenth.

11.1 Space Figures and Cross Sections Unit 9 Understanding 3D Figures

Vocabulary Polyhedron A 3-dimensional figure whose surfaces are polygons Face Each polygon of the polyhedron Edge Segment formed by the intersection of two faces Vertex Point where three or more edges intersect Polyhedron A 3-dimensional figure whose surfaces are polygons Face Each polygon of the polyhedron Edge Segment formed by the intersection of two faces Vertex Point where three or more edges intersect

Vocabulary Eulers Formula The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E). F + V = E + 2 In two dimensions, Eulers Formula reduces to F + V = E + 1. Eulers Formula The sum of the number of faces (F) and vertices (V) of a polyhedron is two more than the number of its edges (E). F + V = E + 2 In two dimensions, Eulers Formula reduces to F + V = E + 1.

Using Eulers Formula What does a net for the doorstop at the right look like? Label the net with its appropriate dimensions. Solution: Draw the net and then verify Eulers Formula. Faces (F) = 5 Vertices (V) = 10 Edges (E) = 14 F + V = E + 1 5 + 10 = 14 + 1 15 = 15 What does a net for the doorstop at the right look like? Label the net with its appropriate dimensions. Solution: Draw the net and then verify Eulers Formula. Faces (F) = 5 Vertices (V) = 10 Edges (E) = 14 F + V = E + 1 5 + 10 = 14 + 1 15 = 15

Your Turn! Draw a net the 3-dimensional figure.

Vocabulary Cross-section Intersection of a solid and a plane Cross-section Intersection of a solid and a plane

Drawing a Cross-section Draw the horizontal cross-section for a triangular prism. Solution: To draw a cross section, visualize a plane intersecting one face at a time in parallel segments. Draw the parallel segments, then join their endpoints and shade the cross section. Draw the horizontal cross-section for a triangular prism. Solution: To draw a cross section, visualize a plane intersecting one face at a time in parallel segments. Draw the parallel segments, then join their endpoints and shade the cross section.

Your Turn! Draw and describe the cross section formed by intersecting the rectangular prism with the plane described. A) a plane that contains the vertical line of symmetry Solution: See board for cross-section; the cross-section is a rectangle B) a plane that contains the horizontal line of symmetry Solution: See board for cross-section; the cross-section is a rectangle Draw and describe the cross section formed by intersecting the rectangular prism with the plane described. A) a plane that contains the vertical line of symmetry Solution: See board for cross-section; the cross-section is a rectangle B) a plane that contains the horizontal line of symmetry Solution: See board for cross-section; the cross-section is a rectangle

11.4 Volumes of Prisms and Cylinders Unit 9 Understanding 3D Figures

Vocabulary Volume The space that a figure occupies; It is measured in cubic units Cavalieris Principle If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume. Volume The space that a figure occupies; It is measured in cubic units Cavalieris Principle If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume.

Vocabulary

Finding the Volume of Rectangular Prisms

Find the Volume of Triangular Prisms

Your Turn! Find the volume of each object. Round to the nearest tenth.

Vocabulary

Finding the Volume of a Cylinder

Your Turn! Find the volume of each figure. the cylindrical part of the measuring cup

Vocabulary Composite Space Figure A 3-dimensional figure that is the combination of two or more simpler figures. Composite Space Figure A 3-dimensional figure that is the combination of two or more simpler figures.

Finding the Volume of a Composite Figure

Your Turn! Find the volume of each composite figure to the nearest tenth.

11.5 Volumes of Pyramids and Cones Unit 9 Understanding 3D Figures

Vocabulary

Finding Volume of a Pyramid

Your Turn! Find the volume of each pyramid. Round to the nearest tenth.

Vocabulary

Find the Volume of a Cone

Your Turn! Find the volume of each figure. Round answers to the nearest tenth.

11.6 Surface Area and Volumes of Spheres Unit 9 Understanding 3D Figures

Vocabulary

Finding the Volume of a Sphere

Your Turn! Find the volume and surface area of a sphere with the given radius or diameter. Round your answers to the nearest tenth.

Your Turn! A sphere has the given volume. Find its radius to the nearest tenth. A) 1436.8 mi 3 B) 808 cm 3 C) 72 m 3 A sphere has the given volume. Find its radius to the nearest tenth. A) 1436.8 mi 3 B) 808 cm 3 C) 72 m 3

Your Turn! The sphere at the right fits snugly inside a cube with 18 cm edges. What is the volume of the sphere? Leave your answers in terms of . The sphere at the right fits snugly inside a cube with 18 cm edges. What is the volume of the sphere? Leave your answers in terms of .

11.7 Areas and Volumes of Similar Solids Unit 9 Understanding 3D Figures

Vocabulary Similar Solids Have the same shape, and all corresponding dimensions are proportional. Similar Solids Have the same shape, and all corresponding dimensions are proportional.

Identifying Similar Solids

Your Turn! Are the given pairs of figures similar?

Vocabulary Volumes of Similar Solids If the scale factor of two similar solids is a:b, then the ratio of their volumes is a 3 :b 3. Volumes of Similar Solids If the scale factor of two similar solids is a:b, then the ratio of their volumes is a 3 :b 3.

Finding the Scale Factor

Your Turn! Each pair of figures is similar. Use the given information to find the scale factor of the smaller figure to the larger figure. Two cubes have sides of length 4 cm and 5 cm. Find the ratio of volumes.

Unit 9 Understanding 3D Figures Geometry 2014-15.

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Transcript of Unit 9 Understanding 3D Figures Geometry 2014-15.