Transcript Chapter10
Polygons and Area § 10.1 Naming Polygons § 10.2 Diagonals and Angle Measure § 10.3 Areas of Polygons § 10.4 Areas of Triangles and Trapezoids § 10.5 Areas of Regular Polygons § 10.6 Symmetry § 10.7 Tessellations Naming Polygons You will learn to name polygons according to the number of sides and ______. angles _____ 1) regular polygon 2) convex 3) concave Naming Polygons closed figure in a plane formed by segments, called sides. A polygon is a _____________ sides or ______. angles A polygon is named by the number of its _____ tri means three. A triangle is a polygon with three sides. The prefix ___ Naming Polygons Prefixes are also used to name other polygons. Prefix Number of Sides Name of Polygon tri- 3 triangle quadri- 4 quadrilateral penta- 5 pentagon hexa- 6 hexagon hepta- 7 heptagon octa- 8 octagon nona- 9 nonagon deca- 10 decagon Naming Polygons A vertex is the point of intersection of two sides. Consecutive vertices are the two endpoints of any side. Q P R U A segment whose endpoints are nonconsecutive vertices is a diagonal. T S Sides that share a vertex are called consecutive sides. Naming Polygons sides congruent. An equilateral polygon has all _____ angles congruent. An equiangular polygon has all ______ equilateral and ___________. equiangular A regular polygon is both ___________ equilateral but not equiangular equiangular but not equilateral regular, both equilateral and equiangular Investigation: As the number of sides of a series of regular polygons increases, what do you notice about the shape of the polygons? Naming Polygons A polygon can also be classified as convex or concave. If all of the diagonals lie in the interior of the figure, then the convex polygon is ______. If any part of a diagonal lies outside of the figure, then the concave polygon is _______. Naming Polygons Diagonals and Angle Measure You will learn to find measures of interior and exterior angles of polygons. Nothing New! Diagonals and Angle Measure Make a table like the one below. 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides quadrilateral 4 Number of Diagonals from One Vertex 1 Number of Triangles 2 Sum of Interior Angles 2(180) = 360 Diagonals and Angle Measure 1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 Diagonals and Angle Measure 1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 Diagonals and Angle Measure 1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900 Diagonals and Angle Measure 1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360 pentagon 5 2 3 3(180) = 540 hexagon 6 3 4 4(180) = 720 heptagon 7 4 5 5(180) = 900 n-gon n n-3 n-2 (n – 2)180 If a convex polygon has n sides, then the sum of the Theorem 10-1 measure of its interior angles is (n – 2)180. Diagonals and Angle Measure In §7.2 we identified exterior angles of triangles. Likewise, you can extend the sides of any convex polygon to form exterior angles. 57° The figure suggests a method for finding the sum of the measures of the exterior angles of a convex polygon. 48° 72° 54° When you extend n sides of a polygon, n linear pairs of angles are formed. The sum of the angle measures in each linear pair is 180. sum of measure of exterior angles sum of measure of exterior angles = sum of measures of linear pairs – sum of measures of interior angles = = n•180 180n – – 180(n – 2) 180n + 360 = 360 74° 55° Diagonals and Angle Measure In any convex polygon, the sum of the measures of the Theorem 10-2 exterior angles, (one at each vertex), is 360. Java Applet Diagonals and Angle Measure Areas of Polygons You will learn to calculate and estimate the areas of polygons. 1) polygonal region 2) composite figure 3) irregular figure Areas of Polygons polygonal region Any polygon and its interior are called a ______________. In lesson 1-6, you found the areas of rectangles. For any polygon and a given unit of measure, there is a Postulate 10-1 unique number A called the measure of the area of the Area Postulate polygon. Area can be used to describe, compare, and contrast polygons. The two polygons below are congruent. How do the areas of these polygons compare? They are the same. Postulate 10-2 Congruent polygons have equal areas. Areas of Polygons composite figures The figures above are examples of ________________. They are each made from a rectangle and a triangle that have been placed together. You can use what you know about the pieces to gain information about the figure made from them. You can find the area of any polygon by dividing the original region into rectangles squares __________, smaller and simpler polygon regions, like _______, triangles and ________. adding the The area of the original polygonal region can then be found by __________ areas of the smaller polygons _________________________. Areas of Polygons The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon. Postulate 10-3 Area Addition Postulate 1 2 3 AreaTotal = A1 + A2 + A3 Areas of Polygons Find the area of the polygon in square units. Area of polygon = Area of Square 3u X 3u = 9u2 Area of Rectangle 1u X 2u = 2u2 = 7u2 3 units 3 units Area of Area of Square Rectangle Areas of Polygons Areas of Triangles and Trapezoids You will learn to find the areas of triangles and trapezoids. Nothing new! Areas of Triangles and Trapezoids Look at the rectangle below. Its area is bh square units. congruent triangles The diagonal divides the rectangle into two _________________. 1 The area of each triangle is half the area of the rectangle, or bh. 2 This result is true of all triangles and is formally stated in Theorem 10-3. h b Areas of Triangles and Trapezoids Consider the area of this rectangle Height A(rectangle) = bh Base A(Triangle ) bh 2 Areas of Triangles and Trapezoids If a triangle has an area of A square units, a base of b units, and a corresponding altitude of h units, then 1 A bh 2 Theorem 10-3 Area of a Triangle h b Areas of Triangles and Trapezoids Find the area of each triangle: 1 4 yd 3 6 yd A = 13 yd2 18 mi 23 mi A = 207 mi2 Because thewill opposite of aof parallelogram have the same length,to first Next we look atsides the area trapezoids. However, it is helpful rectangle the understand area of a parallelogram is closely related to the area of a ________. parallelograms. height base height The area of a parallelogram is found by multiplying the base ____ and the ______. Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base. Areas of Triangles and Trapezoids Starting with a single trapezoid. The height is labeled h, and the bases are labeled b1 and b2 Construct a congruent trapezoid and arrange it so that a pair of congruent legs are adjacent. b1 b2 h b1 b2 The new, composite figure is a parallelogram. It’s base is (b1 + b2) and it’s height is the same as the original trapezoid. The area of the parallelogram is calculated by multiplying the base X height. A(parallelogram) = h(b1 + b2) The area of the trapezoid is one-half of the parallelogram’s area. 1 ATrapezoid hb1 b2 2 Areas of Triangles and Trapezoids If a trapezoid has an area of A square units, bases of b1 and b2 units, and an altitude of h units, then 1 A hb1 b2 2 Theorem 10-4 Area of a Trapezoid b1 h b2 Areas of Triangles and Trapezoids Find the area of the trapezoid: 20 in 18 in 38 in A = 522 in2 Areas of Triangles and Trapezoids Areas of Regular Polygons You will learn to find the areas of regular polygons. 1) center 2) apothem Areas of Regular Polygons Every regular polygon has a ______, center a point in the interior that is equidistant from all the vertices. A segment drawn from the center that is perpendicular to a side of the regular apothem polygon is called an ________. congruent In any regular polygon, all apothems are _________. Areas of Regular Polygons Now, create by drawing segments from the center each vertex on Now The figure multiply shows timesis the a center number and ofall triangles vertices that of amake regular uptopentagon. the regular area ofbelow athis triangle calculated with the following formula: perpendicular to a side. An apothem drawn from the center, and is _____________ either side ofisthe apothem. polygon. 72 There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 5 1 sa 2 72° 72° 72° 72° 72° a s What measure does 5s represent? perimeter Rewrite the formula for the area of a pentagon using P for perimeter. 1 aP 2 Areas of Regular Polygons If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then 1 A aP 2 Theorem 10-5 Area of a Regular Polygon P Areas of Regular Polygons Find the area of the shaded region in the regular polygon. Area of polygon A 1 aP 2 Area of triangle A 1 bh 2 1 A 5.5 ft 40 ft 2 1 A 5.5 ft8 ft 2 A 5.5 ft 20 ft A 5.5 ft 4 ft A 110ft 2 A 22 ft 2 8 ft 5.5 ft triangle To find the area of the shaded region, subtract the area of the _______ pentagon from the area of the ________: The area of the shaded region: 110 ft2 – 22 ft2 = 88 ft2 Areas of Regular Polygons Find the area of the shaded region in the regular polygon. Area of polygon A 1 aP 2 1 A 6.9m48m 2 Area of triangle A 1 bh 2 1 A 8m13.8m 2 A 6.9m 24m A 4m 13.8m A 165.6m 2 A 55.2m 2 6.9 m triangle To find the area of the shaded region, subtract the area of the _______ hexagon from the area of the ________: The area of the shaded region: 165.6 m2 – 55.2 m2 = 110.4 m2 8m Areas of Regular Polygons Section Title You will learn to 1) Section Title Section Title You will learn to 1) Section Title