Postulate 17 The area of a square is the square of the length of a. Postulate 18 If two figures are congruent, then they have the same.


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1 Chapter 11: Areas of Plane Figures (page 422) 111: Areas of Rectangles (page 423) Rectangle Rectangular Region Area is measured in units. Postulate 17 The area of a square is the square of the length of a. s Postulate 18 If two figures are congruent, then they have the same. Postulate 19 The area of a region is the of the areas of its nonoverlapping parts. Any side of a rectangle (parallelogram) can be considered to be a ( ). Any segment perpendicular to the line containing the base from any point on the opposite side is an. The length of an altitude is called the ( ). Theorem 111 The area of a rectangle equals the of its base and height. h b
2 Examples: (1) The area of a square is 9 sq. cm. Find its perimeter. P = (2) The perimeter of a rectangle is 20 cm. If its height is 4 cm, find its area. (3) Consecutive sides of the figure are perpendicular. Find its area Assignment: Written Exercises, Pages 426 & 427: 131 odd # s, 32, 34
3 112: Areas of Parallelograms, Triangles, and Rhombuses (page 429) Any side of a parallelogram may be considered a ( ). Any segment perpendicular to the line containing a base from any point on the opposite side is an. The length of an altitude is called the ( ), of the parallelogram. Theorem 112 The area of a parallelogram equals the product of a base and the height to that base. Theorem 113 The area of a triangle equals the product of a base and the height to that base. Hero s (or Heron s) Formula: a triangle area formula in terms of the sides of the triangle. A s = B C Theorem 114 The area of a rhombus equals the product of its diagonals.
4 Examples: (1) Find the area of a parallelogram with base 12 and height 4. (2) Find the area of the parallelogram. 15 m 10 m 60º (3) Find the area of the triangle (4) Find the area of the rhombus. 5 4 Assignment: Written Exercises, Pages 431 & 432: 123 odd # s
5 113: Areas of Trapezoids (page 435) In a trapezoid, the bases are the sides. The altitude of a trapezoid is any segment from a point on the opposite base. to a line containing one base In a trapezoid, all altitudes have the same, called the ( ). Theorem 115 The area of a trapezoid equals the product of the height and the sum of the bases. The median of a trapezoid is the segment connecting the of the legs. The formula for the length of the median is:. Therefore, another area formula for a trapezoid is: =.
6 Examples: (1) Find the area of the trapezoid (2) Find the area of the trapezoid h 60º x 11 (3) Find the length of the median and the area of the trapezoid that has bases 18 & 24 and height 16. m = (4) If the area of the trapezoid is 128 u 2 and its bases are 12 & 20, find the height. h = Assignment: Written Exercises, Pages 436 & 437: 123 odd # s, * Bonus #30 *
7 114: Areas of Regular Polygons (page 440) Student Activity Using a compass, construct a circle with a given radius. Mark off congruent arcs with chords equal to the radius of the circle. Connect the consecutive points on the circle. The figure formed is a. A regular polygon is both and. Any regular polygon can be inscribed in a. Many of the terms associated with circles are also used with regular polygons. The The A consecutive vertices. of a regular polygon is the center of the circumscribed circle. of the regular polygon is the distance from the center to a vertex. is an angle formed by two radii drawn to two The measure of a central angle of a regular polygon with n sides is. The of a regular polygon is the distance from the center to a side.
8 Theorem 116 The area of a regular polygon is equal to the product of the apothem and the perimeter. explanation of proof: Common Regular Polygons to Know (1) Regular Triangle a.k.a.:
9 (2) Regular Quadrilateral a.k.a.: (3) Regular Hexagon
10 Examples: (1) Find the perimeter and area of a regular triangle with its apothem equal to 9 feet. p = (2) Find the apothem and radius of a regular quadrilateral with an area of 100 sq.yd. a = r = (3) Find the area of a regular hexagon with a side equal to 12. Assignment: Written Exercises, Pages 443 & 444: 115 odd # s, *1820* Worksheet on Lessons 111 to 114: Areas of Polygons
11 115: Circumferences and Areas of Circles (page 445) Note The perimeter of a polygon is defined as the sum of the lengths of the segments making up its sides. A circle is not made up of line segments and therefore, the perimeter of a circle must be defined differently. Student Activity Look at a sequence of regular polygons. Imagine more and more regular polygons having more and more sides. Look at the perimeters of the polygons. The perimeter is approaching the distance around a circle that is about the polygon. This is defined to be the, ( ) of the circle. The ratio of the circumference to the diameter is in all circles. This constant is denoted by the Greek letter, (pi), which represents the exact value for this ratio. Therefore: C = = The area of a circle is defined in a similar way. The areas of the inscribed regular polygons get closer and closer to a This limit is defined to be the of the circle. number. As the regular polygon gets closer to looking like a, the length of the apothem approaches the length of the a and p. Substitute for a and p into the area formula for a regular polygon: 1 / 2 a p Therefore, the area formula for a circle is:
12 Note Since π is an irrational number, there isn t any decimal or fraction that expresses π exactly. Some approximated values for π are:,,, and. Examples: (1) Find the circumference and area of a circle with its radius equal to 8 3. C = (2) Find the circumference of a circle if the area is 100π sq.u. C = (3) A circular garden has a radius of 14 feet. What is the area of the garden? Use π = 22/7. answer: (4) A bicycle wheel has a diameter of 60 cm. How far will it travel if it makes 50 revolutions? Use π = answer: cm (5) The earth has a diameter of approximately 7,913 miles. What is its circumference? answer: Assignment: Written Exercises, Pages 448 to 450: 121 odd # s, * Bonus *
13 116: Arc Lengths and Areas of Sectors (page 452) Two different numbers that describe the size of an arc are. A B (1) the measure of the arc, ie. mab!. O (2) the arc length, which is the of a piece of the circumference. The arc length is a fraction of the whole circumference. If mab! = x, then the length of AB! = The length of the arc equals the fraction times the circumference of the circle. A of a circle is a region bounded by two radii and an arc of the circle. The area of a sector is a fraction of the area of a whole circle. If mab! = x, then the area of sector AOB = The area of the sector equals the fraction times the area of the circle. Examples: (1) In! O with radius 6 and m!aob = 150º, find the lengths of AB " and ACD #. A O B 10º D length of AB! = length of ACD! = C
14 (2) Find the area of the shaded sector in the circle with radius equal to º (3) Find the area of the shaded region in the circle with radius equal to º (4) Find the area of the shaded region in the circle with radius equal to 2 inches. 300º Assignment: Written Exercises, Pages 453 & 454: 119 odd # s
15 117: Ratios of Areas (page 456) 4 3 Comparing Areas of Triangles  refer to classroom exercises on page 8 458, #1 and 2. (1) If two triangles have equal heights, then the ratio of their areas equals the ratio of their. 9 (2) If two triangles have equal bases, then the ratio of their areas equals the ratio of their. (3) If two triangles are similar, then the ratio of their areas equals the square of their. Theorem 117 If the scale factor of two similar figures is a : b, then: (1) the ratio of the perimeters is. (2) the ratio of the areas is.
16 Examples: (1) The scale factor of two similar figures is 3:5. Find the ratio of the perimeters and the ratio of the areas. ratio of perimeters = ratio of areas = (2) The ratio of the areas of two similar figures is 1:4. Find the ratio of their perimeters. ratio of perimeters = (3) The areas of two circles are 36 and 64. Find the ratios of the diameters and circumferences. ratio of diameters = ratio of circumferences = (4) The scale factor of two quadrilaterals is 3:5. The area of the smaller quadrilateral is 27 in 2. Find the area of the larger quadrilateral. A of larger quadrilateral = Assignment: Written Exercises, Pages 458 & 459: 115 odd # s Prepare for Test on Chapter 11: Areas of Plane Figures
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