Download Geometry Day 2

MSIS Geometry and Measurement Day 2 1 Geometry •  The word geometry comes from Greek words meaning “to measure the Earth” •  Basically, Geometry is the study of shapes and is one of the oldest branches of mathemaDcs The Elements •  Knowing that you can’t define everything and that you can’t prove everything, Euclid began by staDng three undefined terms: Ø Point is that which has no part Ø (Straight) Line is a line that lies evenly with the points on itself Ø Plane (Surface) is a plane that lies evenly with the straight lines on itself Actually, Euclid did aOempt to define these basic terms . . . Agenda Day 2 •  8:30 Geometry and Measurement 4 •  9:00 Geometry and Measurement 5 •  10:15 Break 12:30 Lunch •  1:30 Geometry and Measurement 6 •  2:15 Break •  2:30 Geometry and Measurement 6 •  4:00 Departure Visual DescripDve AnalyDc RelaDonal Abstract Dedcut •  Students recognize shapes. •  A rectangle “looks like a door”. •  Students perceive properDes of shapes. •  A rectangle has four sides, all of its sides are straight, opposite sides have equal length. •  Students characterize shapes by their properDes. •  A rectangle has opposite sides of equal length and four right angles. •  Students understand that a rectangle is a parallelogram because it has all the properDes of parallelograms. Content contained is licensed under a CreaDve Commons AOribuDon-­‐ShareAlike 3.0 Unported License Geometry Learning Progression 3
Area Defini*ons Frac*ons 4
Parallel, Perpendicular, Right, Acute, Obtuse angles, Line segments, Ray, Symmetry 5
Volume Coordinate System Categorize Shapes Geometry Learning Progression 3
Area Defini*ons Frac*ons 4
Parallel, Perpendicular, Right, Acute, Obtuse angles, Line segments, Ray, Symmetry 5
Volume Coordinate System Categorize Shapes Grade 3 •  Four CriDcal Areas –  Developing understanding of •  MulDplicaDon/division and strategies within 100 •  FracDons – especially unit fracDons •  THE STRUCTURE OF RECTANGULAR ARRAYS AND OF AREA •  TWO-­‐DIMENSIONAL SHAPES Grade 4 •  Three CriDcal Areas –  Developing understanding of •  GEOMETRIC FIGURES –  CLASSIFYING BY PROPERTIES •  MulD-­‐digit mulDplicaDon/division –  Fluency •  FracDons –  Equivalence –  +/-­‐ of unlike fracDons –  MulDplicaDon of whole numbers by fracDons Basic Terms & DefiniDons •  A ray starts at a point (called the endpoint) and extends indefinitely in one direcDon. A B AB •  A line segment is part of a line and has two endpoints. A B AB •  An angle is formed by two rays with the same endpoint. side vertex side •  An angle is measured in degrees. The angle formed by a circle has a measure of 360 degrees. •  A right angle has a measure of 90 degrees. •  A straight angle has a measure of 180 degrees. •  A simple closed curve is a curve that we can trace without going over any point more than once while beginning and ending at the same point. •  A polygon is a simple closed curve composed of at least three line segments, called sides. The point at which two sides meet is called a vertex. •  A regular polygon is a polygon with sides of equal length. Polygons # of sides 3 4 5 6 7 8 9 10 name of Polygon triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon Quadrilaterals •  Recall: a quadrilateral is a 4-­‐sided polygon. We can further classify quadrilaterals: q A trapezoid is a quadrilateral with at least one pair of parallel sides. q A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. q A kite is a quadrilateral in which two pairs of adjacent sides are congruent. q A rhombus is a quadrilateral in which all sides are congruent. q A rectangle is a quadrilateral in which all angles are congruent (90 degrees) q A square is a quadrilateral in which all four sides are congruent and all four angles are congruent. From General to Specific More specific Quadrilateral trapezoid kite parallelogram rhombus rectangle square Perimeter and Area •  The perimeter of a plane geometric figure is a measure of the distance around the figure. •  The area of a plane geometric figure is the amount of surface in a region. area perimeter Rectangle and Square s w l
Perimeter = 2w + 2l Perimeter = 4s Area = lw Area = s2 •  What is the greatest area you can make with a rectangle that has a perimeter of 24 units? Geometric measurement: understand concepts of angle and measure angles. •  4.MD.5 •  Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: •  a. An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fracDon of the circular arc between the points where the two rays intersect the circle. An angle that turns through of a circle is called a "one-­‐
degree angle," and can be used to measure angles. •  b. An angle that turns through n one-­‐degree angles is said to have an angle measure of n degrees. The diagram below will help students understand that an angle measurement is not related to an area since the area between the 2 rays is different for both circles yet the angle measure is the same. Geometric measurement: 4.MD.6 •  Measure angles in whole-­‐number degrees using a protractor. •  Sketch angles of specified measure For each part below, explain how the measure of the unknown angle can be found without using a protractor. Problems •  Joey knows that when a clock’s hands are exactly on 12 and 1, the angle formed by the clock’s hands measures 30°. •  What is the measure of the angle formed when a clock’s hands are exactly on the 12 and 4? Problem solving •  List Dmes on the clock in which the angle between the hour and minute hands is 90°. Use a student clock, watch, or real clock. Verify your work using a protractor. Geometric measurement: 4.MD.7 •  Recognize angle measure as addiDve. When an angle is decomposed into non-­‐overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. •  Solve addiDon and subtracDon problems to find unknown angles on a diagram in real world and mathemaDcal problems, e.g., by using an equaDon with a symbol for the unknown angle measure. If the two rays are perpendicular, what is the value of ? Write equaDons using variables to represent the unknown angle measurements. Find the unknown •  . angle measurements numerically. Grade 5 •  Three CriDcal Areas –  Extend division to 2-­‐digit divisors, integrate decimal fracDons, understand decimals to hundredths, develop fluency with whole numbers and decimals –  UNDERSTANDING OF VOLUME –  FracDons •  Fluency with +/-­‐ •  Understanding of mulDplicaDon •  Division in limited cases Geometry Learning Progression 3
Area Defini*ons Frac*ons 4
Parallel, Perpendicular, Right, Acute, Obtuse angles, Line segments, Ray, Symmetry 5
Volume Coordinate System Categorize Shapes Graph points on the coordinate plane to solve real-­‐
world and mathemaDcal problems. •  5.G.1 – •  Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersecDon of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. •  Understand that the first number indicates how far to travel from the origin in the direcDon of one axis, and the second number indicates how far to travel in the direcDon of the second axis, with the convenDon that the names of the two axes and the coordinates correspond (e.g., x-­‐axis and x-­‐coordinate, y-­‐axis and y-­‐ coordinate). Problem •  Edna drew a square that had an area of 36 square units using the grid shown below. She started drawing her square at (7,10). Draw the square that Edna could have made in the grid below. Be sure to label the coordinates of each vertex of the square. Problem •  (a) Plot the following coordinate pairs on a grid: A (2,4) B (2,8) C (8,4) D(8,8) •  (b) If you connected point A to point B. Then connect point D to point •  C. Describe the relaDonship between the line segments. •  (c) If you connect point A to point B, point B to point D, point D to point C, and point C to point A, what geometric figure have you drawn? •  (d) List two properDes of this geometric figure. Problem solving 5.G.2 •  Represent real world and mathemaDcal problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situaDon. Sample •  Sara has saved $20. She earns $8 for each hour she works. ♣ If Sara saves all of her money, how much will she have aser working 3 hours? 5 hours? 10 hours? ♣ Create a graph that shows the relaDonship between the hours Sara worked and the amount of money she has saved. •  ♣ What other informaDon do you know from analyzing the graph? Classify two-­‐dimensional figures into categories based on their properDes. 5.G.3 •  Understand that aOributes belonging to a category of two-­‐dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. Sample •  If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms. •  ♣ A parallelogram has 4 sides with both sets of opposite sides parallel. What types of quadrilaterals are parallelograms? •  ♣ Regular polygons have all of their sides and angles congruent. Name or draw some regular polygons. •  ♣ All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or false? •  ♣ A trapezoid has 2 sides parallel so it must be a parallelogram. True or false? Classify two-­‐dimensional figures into categories based on their properDes •  5.G.4 – Classify two-­‐dimensional figures in a hierarchy based on properDes. ProperDes of figure may include: • 
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♣ ProperDes of sides – parallel, perpendicular, congruent, number of sides ♣ ProperDes of angles – types of angles, congruent • Examples: ♣ A right triangle can be both scalene and isosceles, but not equilateral. ♣ A scalene triangle can be right, acute, and obtuse. • Triangles can be classified by: ♣ Angles σ Right – the triangle has one angle that measures . σ Acute: The triangle has exactly three angles that measure between σ Obtuse: The triangle has exactly one angle that measures greater than ♣ Sides σ Equilateral: All sides of the triangle are the same length. σ Isosceles: At least two sides of the triangle are the same length. σ Scalene: No sides of the triangle are the same length. Convert like measurement units within a given measurement system. •  5.MD.1 – •  Convert among different-­‐sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving mulD-­‐step, real world problems. Represent and interpret data. 5.MD.2 – Make a line plot to display a data set of measurements in fracDons of a unit (1/2, 1/4, 1/8). Use operaDons on fracDons for this grade to solve problems involving informaDon presented in line plots. For example, given different measurements of liquid in idenAcal beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. Ten beakers, measured in liters, are filled with a liquid. •  The above line plot shows the amount of liquid in liters in 10 beakers. If the liquid is redistributed equally, how much liquid would each beaker have? Geometric measurement: understand concepts of volume and relate volume to mulDplicaDon and to addiDon. 5.MD.3 – •  Recognize volume as an aOribute of solid figures and understand concepts of volume measurement. a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. •  b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. Geometric measurement 5.MD.4 – •  Measure volumes by counDng unit cubes, using cubic cm, cubic in, cubic s, and improvised units. Geometric Measurement 5.MD.5 – Relate volume to the operaDons of mulDplicaDon and addiDon and solve real world and mathemaDcal problems involving volume. • 
Find the volume of a right rectangular prism with whole-­‐number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by mulDplying the edge lengths, equivalently by mulDplying the height by the area of the base. • 
Represent threefold whole-­‐number products as volumes, e.g., to represent the associaDve property of mulDplicaDon. • 
Apply the formulas V = l w h and V = b h for rectangular prisms to find volumes of right rectangular prisms with whole-­‐number edge lengths in the context of solving real world and mathemaDcal problems. • 
Recognize volume as addiDve. Find volumes of solid figures composed of two non-­‐
overlapping right rectangular prisms by adding the volumes of the non-­‐
overlapping parts, applying this technique to solve real world problems. Determine the volume of concrete needed to build the steps in the diagram below. Aaron says more informaDon is needed to find the volume of the prisms. Is Aeron mistaken? Can you calculate the volume of the prisms ? . The tank, shaped like a rectangular prism, is filled to the top with water
•  Will the beaker hold all the water in the box? If yes, how much more will the beaker hold? If not, how much more will the cube hold than the beaker? Explain how you know. Problem •  A rectangular tank with a base area of 24 cm2 is filled with water and oil to a depth of 9 cm. The oil and water separate into two layers when the oil rises to the top. If the thickness of the oil layer is 4 cm, what is the volume of the water? •  Decide whether each of these statements is always, someDmes, or never true. If it is someDmes true, draw and describe a figure for which the statement is true and another figure for which the statement is not true. •  A rhombus is a square •  A triangle is a parallelogram •  A square is a parallelogram •  A square is a rhombus •  A parallelogram is a rectangle •  A trapezoid is a quadrilateral •  Which quadrilateral has a greater area? •  Quadrilateral A has its perimeter equal to 44 units. •  Quadrilateral B has the sum of its interior angles equal to 360 degrees. •  A rectangular prism has a volume of 144 cubic units and a base of 48 square units. What could the possible dimensions be? •  Cari is the lead architect for the city’s new aquarium. All of the tanks in the aquarium will be rectangular prisms where the side lengths are whole numbers. •  Cari’s first tank is 4 feet wide, 8 feet long and 5 feet high. How many cubic feet of water can her tank hold? •  Cari knows that a certain species of fish needs at least 240 cubic feet of water in their tank. Create 3 separate tanks that hold exactly 240 cubic feet of water. •  In the back of the aquarium, Cari realizes that the ceiling is only 10 feet high. She needs to create a tank that can hold exactly 100 cubic feet of water. Name one way that she could build a tank that is not taller than 10 feet. Aaron says more informaDon is needed to find the volume of the prisms. Is Aeron mistaken? Can you calculate the volume of the prisms ? . The tank, shaped like a rectangular prism, is filled to the top with water
•  Will the beaker hold all the water in the box? If yes, how much more will the beaker hold? If not, how much more will the cube hold than the beaker? Explain how you know. Problem •  A rectangular tank with a base area of 24 cm2 is filled with water and oil to a depth of 9 cm. The oil and water separate into two layers when the oil rises to the top. If the thickness of the oil layer is 4 cm, what is the volume of the water? Read Grade 6 Introduction to CC Math
Standards to look for the Geometry
Grade 6 •  Four CriDcal Areas – Connect raDo and rate to whole number mulDplicaDon and division – Division of fracDons and extending to raDonal number system including negaDves – WriDng, interpreDng, and using expressions and equaDons – Develop understanding of staDsDcal thinking Solve real-­‐world and mathemaDcal problems involving area, surface area, and volume. 6.G.1 •  Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; •  Apply these techniques in the context of solving real-­‐world and mathemaDcal problems. Discovering the formula for finding the area of right triangles. 1. Using a ruler, draw a diagonal (from one corner to the opposite corner) on shapes A, B, and C. 2. Along the top edge of shape D, mark a point that is not a vertex. Using a ruler, draw a line from each boOom corner to the point you marked. (Three triangles should be formed.) 3. Cut out the rectangles. Then, divide A, B, and C into two parts by cuxng along the diagonal, and divide D into three parts by cuxng along the lines you drew. What shapes did you get? 4. How do the areas of the triangles compare to the area of the original shape? Calculate the area of each triangle using two different methods. Figures are not drawn to scale. Problems Use Grid paper to a. Find the area of a triangle with a base length of three units and a height of four units. b. Find the area of the trapezoid shown below using the formulas for rectangles and triangles. Another example Problems •  A rectangle measures 3 inches by 4 inches. If the lengths of each side double, what is the effect on the area? •  The area of the rectangular school garden is 24 square units. The length of the garden is 8 units. What is the length of the fence needed to enclose the enDre garden? Problem The 6th grade class at Hernandez School is building a giant wooden H for their school. The H will be 10 feet tall and 10 feet wide and the thickness of the block leOer will be 2.5 feet. The truck that will be used to bring the wood from the lumberyard to the school can only hold a piece of wood that is 60 inches by 60 inches. What pieces of wood (how many pieces and what dimensions) are needed to complete the project? How large will the H be if measured in square feet? Solve real-­‐world and mathemaDcal problems involving area, surface area, and volume 6.G.2 Find the volume of a right rectangular prism with fracDonal edge lengths by packing it with unit cubes of the appropriate unit fracDon edge lengths, and show that the volume is the same as would be found by mulDplying the edge lengths of the prism. Apply the formulas V = lwh and V = bh to find volumes of right rectangular prisms with fracDonal edge lengths in the context of solving real-­‐world and mathemaDcal problems Visualizing and ManipulaDng to understand Volume •  Students need mulDple opportuniDes to measure volume by filling rectangular prisms with blocks and looking at the relaDonship between the total volume and the area of the base. •  Through these experiences, students derive the volume formula (volume equals the area of the base Dmes the height). •  Students can explore the connecDon between filling a box with unit cubes and the volume formula using interacDve applets such as the Cubes Tool on NCTM’s IlluminaDons. Visualizing and ManipulaDng to understand Volume In addiDon to filling boxes, students can draw diagrams to represent fracDonal side lengths, connecDng with mulDplicaDon of fracDons. This process is similar to composing and decomposing two-­‐dimensional shapes. Which prism will hold more 1 in. x 1 in. x 1 in. cubes? How many more cubes will the prism hold? A ¼ in. cube was used to pack the prism. How many ¼ in. cubes will it take to fill the prism? What is the volume of the prism? How is the number of cubes related to the volume? Prac*cal Applica*on Cube-­‐shaped boxes will be loaded into the cargo hold of a truck. The cargo hold of the truck is in the shape of a rectangular prism. The edges of each box measure 2.50 feet and the dimensions of the cargo hold are 7.50 feet by 15.00 feet by 7.50 feet, as shown below. Kelly wants to wrap 20 golf balls, each in a cube-­‐shaped box, together in one larger box. Which arrangement will use the least wrapping paper? •  Build a box with 20 cubes •  Sketch each box, label dimensions, find area of each face and the total surface area •  Display all boxes on chart paper •  Label which arrangement has the largest surface area and which has the smallest. Wait a minute… •  How can the boxes have the same volume of 20 cubes and have different surface areas? •  Discuss with your table group how students in 6th grade may respond to the above quesDon. Problem Sketch and Solve a. Determine the volume of a rectangular prism length and width are in a raDo of 3:1. b. The width and height are in a raDo of 2: 3. The length of the rectangular prism is 5 s. Solve real-­‐world and mathemaDcal problems involving area, surface area, and volume 6.G.3 – Draw polygons in the coordinate plane given coordinates for the verDces; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-­‐world and mathemaDcal problems. PloWng figures on the coordinate plane Determine the area of both polygons on the coordinate plane, and explain why you chose the methods you used. Then write an expression that could be used to determine the area of the figure. 1.  Plot and connect the points A (3, 2), B (3, 7), and C (8, 2). Name the shape and determine the area of the polygon. 2. Plot and connect the points E(-­‐8, 8), F (-­‐2, 5), and G (-­‐7, 2). Then give the best name for the polygon and determine the area. Problem •  On a map, the library is located at (-­‐2, 2) , the city hall building is located at (0, 2) , and the high school is located at (0,0). Represent the locaDons as points on a coordinate grid with a unit of 1 mile. –  What is the distance from the library to the city hall building? The distance from the city hall building to the high school? How do you know? –  What shape is formed by connecDng the three locaDons? The city council is planning to place a city park in this area. How large is the area of the Solve real-­‐world and mathemaDcal problems involving area, surface area, and volume •  6.G.4 •  Represent three-­‐dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. •  Apply these techniques in the context of solving real-­‐world and mathemaDcal problems Examples •  Describe the shapes of the faces needed to construct a rectangular pyramid. Cut out the shapes and create a model. Did your faces work? Why or why not? •  Create the net for a given Prism or pyramid and then use the net to calculate the surface area. Classify each net as represenDng a rectangular prism, a triangular prism, or a pyramid. Reset buOon. What three-­‐dimensional figure will the net create? •  What is the volume of the figure? •  What is the surface area of the figure? A box needs to be painted. How many square inches will need to be painted to cover every surface? Draw and label a net for the following figure. Then use the net to determine the surface area of the figure. Sketch a net of this pizza box. It has a square top that measures 16 inches on a side, and the height is 2 inches. Treat the box as a prism, without counDng the interior flaps that a pizza box usually has. 6.EE.9 •  6.EE.9 – •  Use variables to represent two quanDDes in a real-­‐world problem that change in relaDonship to one another; write an equaDon to express one quanDty, thought of as the dependent variable, in terms of the other quanDty, thought of as the independent variable. •  Analyze the relaDonship between the Each week QuenDn earns $30. If he saves this money, create a graph that shows the total amount of money QuenDn has saved from week 1 through week 8. Write an equaDon that represents the relaDonship between the number of weeks that QuenDn has saved his money, 𝑤𝑤, and the total amount of money in dollars that he has saved, 𝑠𝑠. Then, name the independent and dependent variables. Write a sentence that shows this relaDonship. Using Precise Mathema*cal Language •  Remember that according to van Heile, language is the basis for understanding and communicaDng about geometry. Before we can find out what a student knows we must establish a common language and vocabulary. c a Triangle h b Perimeter = a + b + c Area = 1 bh 2
The height of a triangle is measured perpendicular to the base. Parallelogram a h b Perimeter = 2a + 2b Area = hb à Area of a parallelogram = area of rectangle with width = h and length = b a c Trapezoid b d h b a Perimeter = a + b + c + d 1
Area = h(a + b) 2
à Parallelogram with base (a + b) and height = h with area = h(a + b) à But the trapezoid is half the parallelgram à Ex: Name the polygon 2 1 6 3 5 à hexagon 4 1 2 5 à pentagon 3 4 Ex: What is the perimeter of a triangle with sides of lengths 1.5 cm, 3.4 cm, and 2.7 cm? 1.5 2.7 3.4 Perimeter = a + b + c = 1.5 + 2.7 + 3.4 = 7.6 Ex: The perimeter of a regular pentagon is 35 inches. What is the length of each side? s Recall: a regular polygon is one with congruent sides. Perimeter = 5s 35 = 5s s = 7 inches Ex: A parallelogram has a based of length 3.4 cm. The height measures 5.2 cm. What is the area of the parallelogram? Area = (base)(height) 5.2 3.4 Area = (3.4)(5.2) = 17.86 cm2 Ex: The width of a rectangle is 12 s. If the area is 312 s2, what is the length of the rectangle? 12 312 Area = (Length)(width) Let L = Length L 312 = (L)(12) L = 26 s Check: Area = (Length)(width) = (12)(26) = 312 r d Circle •  A circle is a plane figure in which all points are equidistance from the center. •  The radius, r, is a line segment from the center of the circle to any point on the circle. •  The diameter, d, is the line segment across the circle through the center. d = 2r •  The circumference, C, of a circle is the distance around the circle. C = 2πr •  The area of a circle is A = πr2. Arizona Progressions •  Read through Grade 3 Geometry and Geometric Measurement and jot down a few ideas to share. • 
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Think about what you read. Note important ideas. Share ideas with a partner Share ideas with the whole group. Find the Circumference 1.5 cm •  The circumference, C, of a circle is the distance around the circle. C = 2πr •  C = 2πr •  C = 2π(1.5) •  C = 3π cm Find the Area of the Circle •  The area of a circle is A = πr2 8 in •  d=2r •  8 = 2r •  4 = r •  A = πr2 •  A = π(4)2 •  A = 16π sq. in. Composite Geometric Figures •  Composite Geometric Figures are made from two or more geometric figures. •  Ex: + •  Ex: Composite Figure -­‐ Ex: Find the perimeter of the following composite figure 15 8 = + Rectangle with width = 8 and length = 15 Perimeter of parDal rectangle = 15 + 8 + 15 = 38 Half a circle with diameter = 8 à radius = 4 Circumference of half a circle = (1/2)(2π4) = 4π. Perimeter of composite figure = 38 + 4π. Ex: Find the perimeter of the following composite figure 60 12 28 42 ? = b 12 28 ? = a 60 a b 42 60 = a + 42 à a = 18 28 = b + 12 à b = 16 Perimeter = 28 + 60 + 12 + 42 + b + a = 28 + 60 + 12 + 42 + 16 + 18 = 176 Ex: Find the area of the figure 3 3 3 8 Area of triangle = ½ (8)(3) = 12 8 3 8 Area of rectangle = (8)(3) = 24 Area of figure = area of the triangle + area of the square = 12 + 24 = 36. Ex: Find the area of the figure 4 4 3.5 3.5 Area of rectangle = (4)(3.5) = 14 4 The area of the figure = area of rectangle – cut out area = 14 – 2π square units. Diameter = 4 à radius = 2 Area of circle = π22 = 4π à Area of half the circle = ½ (4π) = 2π
Ex: A walkway 2 m wide surrounds a rectangular plot of grass. The plot is 30 m long and 20 m wide. What is the area of the walkway? 2 30 20 What are the dimensions of the big rectangle (grass and walkway)? Width = 2 + 20 + 2 = 24 Length = 2 + 30 + 2 = 34 2 Therefore, the big rectangle has area = (24)(34) = 816 m2. What are the dimensions of the small rectangle (grass)? 20 by 30 The small rectangle has area = (20)(30) = 600 m2. The area of the walkway is the difference between the big and small rectangles: Area = 816 – 600 = 216 m2. Find the area of the shaded region 10 10 Area of square = 102 = 100
10 Area of each circle = π52 = 25π
r = 5 r = 5 ¼ of the circle cuts into the square. But we have four ¼ 4(¼)(25π ) cuts into the area of the square. Therefore, the area of the shaded region = area of square – area cut out by circles = 100 – 25π square units