Chapter 11- Areas of Plane Figures By Lilli Leight, Zoey Kambour, and Claudio Miro

11.1-Areas of Rectangles  Postulate 17-The area of a square is the square of the length of a side.  A=S Length: 1 unit Area: 1 square unit By counting, Area=9 square units By using the formula, Area=3 2 =9

11.1-Areas of Rectangles  Postulate 18 (Area Congruence Postulate)-If two figures are congruent, then they have the same area.  Postulate 19 (Area Addition Postulate)-The area of a region is the sum of the areas of its non-overlapping parts.

11.1-Areas of Rectangles  Theorem 11.1-The area of a rectangle equals the product of its base and height.  A=bh AH2H2 B2B2 A b h hbhb hbhb Given: A rectangle with base b and height h Prove: A=bh Proof: Draw-the given rectangle with area A, a congruent rectangle with area A, a square with area b 2, a square with area h 2 Area of big square= 2A+b 2 +h 2 Area of big square= (b+h) 2 = b 2 +2bh+h 2 2A+b 2 +h 2 = b 2 +2bh+h62 2A = 2bh A = bh

11.1 Practice Problems  What is the area of a rectangle with a base of 5 and a height of 7?  What is the area of a square with a base of 4 and a height of 4?

11.2-Areas of Parallelograms, Triangles, and Rhombuses  Theorem 11.2-The area of a parallelogram equals the product of a base and the height to that base  A=bh SR QP h V I IIIII b h T Given: PQRS Prove: A=bh Key steps of proof: 1.) Draw altitudes PV and QT, forming two rt, triangles 2.)Area I=Area III 3.) Area of PQRS= Area II + Area I = Area II + Area III = Area of rect. PQTV = bh

11.2 Areas of Parallelograms, Triangles, and Rhombuses  Theorem 11.3-The area of a triangle equals half the product of a base and the height to that base  A=1/2bh h W Z Y b Given: XYZ Prove: A=1/2bh Key steps of proof: 1.) Draw XW parallel to YZ and ZW parallel to YX forming WXYZ 2.) XYZ congruent to ZWX (SAS or SSS) 3.) Area of XYZ = ½ x Area of WXYZ =1/2bh X

11.2-Areas of Parallelograms, Triangles, and Rhombuses  11.4-The area of a rhombus equals half the product of its diagonals.  A=1/2d 1 d 2 Given: Rhombus ABCD with diagonals d 1 and d 2 Prove: A=1/2d 1 d 2 Key steps of proof: 1.) ADC congruent ABC (SSS) 2.)Since DB is perpendicular to AC, the area of ADC = ½ bh = ½ x d 1 x 1/2d 2 =1/4d 1 d 2 3.)Area of rhombus ABCD=2 x 1/4d 1 d 2 = 1/2d 1 d 2 1/2d 2 d1d1 D C BA

11. 2 Examples 1.) Find the area of a parallelogram with sides 8 and 15, and the acute angle equal to 35 degrees. Sin(35)=X/8 X=4.588 x 15 = ) The area of a triangle is 410 with a base of 41. Find its height. A=410 A=1/2bh 410=1/2(41)(h) 410=(20 x 5)(h) h=20

11.2 Practice Problems  Find the area of a parallelogram with sides 6 cm and 8 cm, and a 135 degree angle.  A rhombus has a perimeter of 60 and one diagonal of 24. Find its area.  Find the area of:  1.) An equilateral triangle with a perimeter of 24  2.) An isosceles triangle with sides 13, 13, and 10.  3.) triangle with a hypotenuse of 12 inches.

11.3 Area of a Trapezoid  Theorem 11.5-The area of a trapezoid equals half the product of the height and the sum of the bases.  (A=1/2h(b 1 +b 2 ))  Also A=h(median) A DC B II I b1b1 b2b2 h h Key steps of proof: 1.) Draw a diagonal BD of trap. ABCD, forming two triangular regions, I and II, each with a height of h. 2.) Area of a trapezoid = Area I+ Area II = 1/2b 1 h+1/2b 2 h = 1/2h(b 1 +b 2 )

11.3 Example  Find the area of a trapezoid with a height of 7 and a median of 15.  15 x 7=

11.3 Practice Problem  A trapezoid has an area of 75 cm 2 and a height of 5 cm. How long is the median? 5 cm

11.4 Regular Polygon  A regular Polygon is any convex shape whose sides are all the same length and angles are all the same measure.

11.4 Regular Polygons  Center of a regular polygon= Center of circumscribed circle.  Radius of a regular polygon= distance from a center to a vertex.  Central Angle of a regular polygon= an angle formed by 2 radii drawn to consecutive vertices.  Apothem of a regular polygon = Perpendicular distance from the center to one of the sides of the polygon. Regular Polygons can be inscribed inside of a circle. Using this relationship, some definitions were derived.

11.4 Practice Problems  Find the perimeter and the area of each figure. 1. A regular octagon with sides 4 and apothem a. 2. A regular pentagon side s and apothem of A regular decagon with side s and apothem a. 5 9

11.4 Theorem 11-6  The area of a regular polygon is one half of the product of the apothem and the perimeter.  A=1/2*P*a  P=Perimeter  a= Apothem  Ex:  P=10*6=60  a=8.66  A=(1/2)*60*8.66=259.8 sq. units

11.5 Circumference  Circumference: Perimeter of a circle.  It is found by the product of twice the radius (diameter) and pi.  C= 2πr or C= πd  C= circumference r= radius d= diameter  Ex: Find the circumference of a circle with a radius of 12.  C=? r= 12  C= (2)π(12)= 24π units

11.5 Area  Area= As the area of the inscribed regular polygons get closer and closer to a limiting number defined to be the area of a circle.  It is found by the product of the radius square and pi.  A= πr 2 r = radius  Ex: Find the area of a circle with a radius of 27  A=? r= 27  A= (27 2 ) π = 729π sq. units

11.5 Practice Problems 1. A circle has an area of 18 in. Find the circumference of the rim. 2. Find the area of a circle with a radius of 7 3. Find the radius and area of a circle with a circumference of 20π. 4. Find the radius and circumference of a circle with an area of 25π

11.6 Arc Lengths and Areas of Sectors  Sector- region bounded by two radii and an arc of the circle.

11.6 Finding the length of a sector Length of sector: x/360(2πr)  x = number of degrees in sector  r = radius Ex. Find the length of the sector 120/360(2x9π) 1/3(18 π) = 6 π 120 9

11.6 Finding the area of a sector  Area of sector: x/360 = πr 2  x = degree of sector  r = radius Ex. Find the area of the sector 45/360(4 2 π) 5/72(16 π)= 10 π/9 45 4

11.6 Finding the length of the radii  If you are only given the area or length of a sector and the sector degree measurement, you can still find the length of the radius.  Practice problems 1. Find the length of the radius with length of the sector is 2π and the degree measurement is Find the length of the radius with length of sector being 4π and area of sector 120.

11.6 Practice Problems  Find the area and the length of each sector 1 1

11.7 Ratios of Areas  Theorem 11-7: If the scale factor of two similar figures a:b, then  The ratio of the perimeters is a:b  The ratio of the areas is a 2 :b 2 Ex. Find the ratio of the perimeters and the areas of the two similar figures. The scale factor is 8:12 or 2:3. Therefore, the ratio of the perimeter is 2:3. The ratio of the areas is 2 3 :3 2 or 4:9. 812

11.7 Practice Problems  If the scale factor is 1:4, what is the ratio of the perimeter and the ratio of the areas?  If the ratio of the areas I 25:1, what is the ratio of the perimeter and the scale factor?  A quadrilateral with sides 8cm, 9cm, 6cm and 5cm has area 45 cm 2. Find the area of a similar quadrilateral who's longest side is 15 cm.