Measurement 1 PYTHAGOREAN THEOREM. The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of
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1 Measurement 1 PYTHAGOREAN THEOREM Remember the Pythagorean Theorem: The area of the square on the hypotenuse of a right triangle is equal to the sum of the areas of the squares on the other two sides a b c or c a b c b a Example 1: Calculate the unknown side of the following triangles. a) b) PERIMETER A is a closed figure formed by 3 or more line segments.
2 Example 1: Examples of different polygons. Hexagon 6 sides Octagon 8 sides The distance around a polygon is called the. Example 2: Find the distance around the following polygons: c) 7 m d) 5 m 6 m 5 cm 4 cm 5 cm 7 m 6 m 6 cm Using the paper provided construct 2 different figures with a perimeter of. How many regular polygons can be constricted with sides that are whole numbers greater than 3 and whose perimeters are 48? Draw them. Compare to your group members. Are they all the same? The perimeter of a circle is called the. The is approximately 3.14 times the diameter of the circle. We represent this constant, approximately 3.14, by the symbol (read pi ). There is a button on your calculator for the constant,. You can also use the 3.14 decimal approximation only if the button is not on your calculator. The circumference of a circle is represented by: C d or C (2 r) 2 r
3 Example 1: Calculate the circumference of the following circles. a) b) 10 cm 2.5 m AREA OF A RECTANGLE AND SQUARE is the number of square units needed to cover a surface. To calculate the area of a square or rectangle: A lw (or (multiply the length and the width to get the area) 2 A s s s for the special case of a square) Example 1: Find the area of each figure. c) 10 cm 15 cm d) A square with side length 6 cm. Example 2: Find the length of the base of a rectangle is the area is 98 m 2 and the height is 7 m.
4 AREA OF A PARALLELOGRAM A is a quadrilateral (polygon with 4 sides) with opposite sides parallel and equal in length. (Note: parallel lines never meet.) A parallelogram has the same area as a rectangle where the base (b) equals the length (l) and the height (h) equals the width (w). h b To calculate the area of a parallelogram: A bh (multiply the base and the height to get the area) Example 3: Find the area of the following parallelogram. 15 mm 6 mm 12 mm AREA OF A TRIANGLE h b To calculate the area of a triangle: bh A or 2 A 1 2 bh (multiply the base and the height and then divide by 2 to get the area)
5 Why do we divide by two? Example 1: Find the area of the following triangle. 10 cm 6.6 cm 8 cm 12 cm AREA OF A CIRCLE Consider the following: What does the figure on the right look like? How do I find the area of that shape? The formula for finding the area of a circle is Example 1: Find the area of a circle with radius 4 cm. Example 2: Given the area of a circle to be 25 cm, find a) the radius and b) the diameter.
6 AREAS OF COMPOSITE FIGURES How can we determine the area of composite figures? We could Example 1: a) Calculate the area of paving stone needed for the patio shown. b) Paving stones cost $42/m 2. How much does it cost to pave the patio? Example 2: Ray s Restaurant has a rectangular patio 15.2 m by 5.1 m. There are 4 small trees on the patio. Each is in a circular pot with a diameter of 1m. What area of the patio is available for seating?
7 Example 3: Cans of juice are packed in cases of 24, in 4-by-6 arrays. The diameter of each can is 6.5 cm. The problem is to find how much of the base of the case is not covered by cans. e) If the cans are touching, what is a. the length of the case? b. the width of the case? f) What is the area of the base of the case? g) What is the area of the base of a can? h) What is the total area of all the bases of all the cans? i) How much of the base of the case is not covered by cans? 8.1 THREE-DIMENSIONAL SOLIDS A is a three dimensional figure with faces that are polygons. Example 1: Examples of geometric solids. cube rectangular prism triangular prism pentagonal prism
8 hexagonal prism square pyramid triangular pyramid pentagonal pyramid cylinder sphere Cone dodecahedron A solid is a 3-dimentional object whose interior is completely. A shell is a 3-dimentional object whose interior is completely. A is a representation of the edges of a polyhedron. Solid Shell Skeleton In a polyhedron, the flat surfaces are called the. The line segment where 2 faces meet is called an. The point (or corner) at which the edges meet is called a. Example 2: Show the base, a face, an edge and a vertex for each of the following solids. NETS A net is a two-dimensional figure that can be folded into a three-dimensional object. Which of the nets below will form a cube? (Hint: There are exactly eleven nets that will form a cube.)
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10 SURFACE AREA A is a three-dimensional figure with faces that are polygons. Label the faces that you see of the following polyhedron: How many faces are there? Draw a diagram (net) that shows all of the faces: Let's calculate the diagram's area. (In 3-dimensional figures, this is called surface area.) Example 1: Find the surface area of each of the following solids: a) Face Work Area Answer:
11 b) Face Work Area 5m 4m 8m 6m Answer: VOLUME The of a solid is defined to be the number of cubic units contained in a space. This is a cubic unit. What is the volume of the following (imagine the cubes are touching): What about this one? Is this figure any different than the one above? 2 cm 4 cm 3 cm
12 What is a quick way of finding the volume? To find the volume (of a prism), multiply the with the. Example 1: Find the volume of each of the following solids: a) Answer: b) 5m 4m 8m 6m Answer:
13 SURFACE AREA AND VOLUME OF A CYLINDER Remember Volume = area of the base height r h Example 1: Find the volume of the following cylinder. 4 cm 10 cm When dealing with surface area, it is often helpful to imagine the figure cut apart, i.e. draw the net. r r h Example 2: Find the surface area of the following cylinder. 4 cm r 10 cm
14 Example 3: A juice can is 16 cm tall and has a diameter of 9 cm. The ends of the can are tin and the body is cardboard. a) What is the volume of the can? 9cm 16 cm b) What area of tin is used? c) What area of cardboard is used? d) What is the total surface area? REPRESENTING 3D SOLIDS When representing a three-dimensional object, there are three views to consider, and. These views are created from the perspective, or point of view, of someone looking at the object.
15 Example 1: Consider the following block object. Its three-view drawings are: Example 2: Consider the following 3-D diagram. Which of the following views matches the front view of the 3-D diagram above? Example 3: The front view matches the 3-D diagram. True or False? Example 4: The three-view drawing on the right is a correct representation of the object on the left. True or False?
16 Example 5: Draw the three views of the following object. Drawing Views of Cube Structures Example 1: Use the model given to you by Ms. Withers try to find a way to draw the top, front, and side views of the figure. Figure Top Front Right Side Left Side A B C Example 2: Determine the number of different models you can build from the following views.
17 Example 3: Draw an isometric sketch of each of the 5 figures on the table. A B C D E F ISOMETRIC VIEWS We have been drawing three-dimensional objects in two dimensions by drawing the three views top, front and side. Example 1: Draw a cube. One perspective: Another perspective:
18 Example 2: Draw the following figures on isometric dot paper. j) k) l) m) Example 3: Sketch the front, top, and side views of the following objects drawn on isometric dot paper. n) o) p) q) Example 4: Consider the top, front, and side views shown below. Build the three-dimensional object which has these views. Using isometric dot paper draw the three-dimensional object.
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