Study Guide 3 Circles, Central and Inscribed angles 1. How do we define a circle? What is a radius, diameter, chord, secant and tangent? What is a minor and major arc? What is a semi-circle? 2. Circular Geoboard (with 24 pegs): a) Without a protractor, construct the following central angles: 75°, 90°, 135°, 180°, 270°. b) Without a protractor, construct the following inscribed angles: 40°, 90°, 135°. c) Classify all angles you drew in a) and b) . (Using words acute, obtuse, reflex, etc.) 3. Circular Geoboard (with 24 pegs): Find the measures of given central and inscribed angles without using a protractor. (See pg. 259, # 8 a) 4. When is the measure of an inscribed angle equal to 90°? Use this knowledge to find the center of a given circle. Solids Classification, Euler’s Formula You should be familiar with the basic terminology and names: Vertex, Edge, Face; Solid, Polyhedron, Prism, Pyramid, Platonic Solid, Regular Polyhedron, Archimedean Solid, Semiregular Polyhedron, Sphere, Cone, and Cylinder . You don’t have to memorize any definitions – the focus is on understanding, see the examples of possible test questions below. A. Name the solids depicted in the following picture. Be as specific as you can (don’t just state “pyramid”; specify what kind of pyramid it is). Sort out the solids using categories polyhedron , prism , pyramid and platonic solid . (Note: Some do not belong to any category and some may belong to more than one group). B. You should be able to represent relationships between/among different categories of solids in Venn Diagrams (in the class we used a tree diagram to classify solids). Example: Draw Venn diagrams to represent the following solids: (3 examples follow so you should have 3 different Venn diagrams) • Solid, polyhedron, pyramid, triangular pyramid, hexagonal prism • Solid, polyhedron, cone, cylinder, rectangular pyramid, triangular prism • Solid, polyhedron, sphere, Platonic Solids, dodecahedron, pentagonal prism, square pyramid, icosahedron. Etc… C. Sketch a cube. Sketch a triangular prism. Sketch a square pyramid. Sketch a cone and cylinder. D. Name all Platonic Solids and describe them (give number of faces and the shape of the faces). E. A semiregular polyhedron called truncated icosahedron (a soccer ball, left) has 32 faces and 60 vertices. How many edges does it have? Another semiregular polyhedron, truncated cube (right) has 14 faces and 36 edges. How many vertices does it have? What did you use to find the answer? Is there a way to find the numbers without counting them on the picture? F. Are the depicted solids regular, semiregular, or none of the above? Explain. (All faces in the depicted solids are regular polygons: equilateral triangles and squares) Volume and Surface Area A. Explain • Why the volume of a prism (not necessarily rectangular) can be calculated as the area of the base x height . • Using the above observation, explain why the volume of a rectangular prism can be calculated as length x width x height . B. Explain, why the surface area of a prism can be calculated as 2x Area of the base + Perimeter of the base x height (of the prism). C. How is the volume of a pyramid related to the volume of a prism with the same base and height? D. Explain where does the formula for the volume of the cone (V = (1/3) ⋅π⋅r2⋅h) come from. E. Derive the formula for the volume and surface area of a cylinder. F. Use the Archimedes’ discovery about the sphere inscribed in a cylinder to derive the formula for the volume and surface area of a sphere. 7. You should be able to find volumes and surface areas of the most common solids. Examples are below and on the next page. Name the following solids and find their volume: L L h = 10cm, L = W = 8 cm h = 10cm, r = 4 cm h = 8cm, l = 10, w = 18 cm h = 10cm, r = 4 cm Find the surface area of the second, third and last solid. See also the solids on the following page: Transformations, Symmetry and Tessellations 1. You should be able to identify isometry, if given an image and its pre -image. Example shows 5 pairs, the pre-image is red. You may use a transparency. 2. You should be able to identify isometry (isometries) if given a portion of a tessellation. Example: Identify isometries which map one shape onto its neighbors. If there is a rotation, find its angle. You may use a transparency. 3. Are the following tessellations regular? Can they be semi-regular? Explain. 4. Draw an irregular figure (it does not have to be a polygon) which tessellates. Explain how you went about creating your figure . Show your tessellation. Motion Geometry 1. Find all three images of the given gray quadrilateral under the following three transformations: a. Line Reflection (the reflection line is labeled “Reflection) b. Rotation (90° clockwise, about the yellow center labeled “rotation”) c. Translation (Direction and distance given by the arrow labeled “Translation”) .