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## Polygons and Area

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**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**Vocabulary**Naming Polygons What You'll Learn You will learn to name polygons according to the number of _____ and ______. sides angles 1) regular polygon 2) convex 3) concave**Naming Polygons**closed figure A polygon is a _____________ in a plane formed by segments, called sides. sides angles A polygon is named by the number of its _____ or ______. A triangle is a polygon with three sides. The prefix ___ means three. tri**Naming Polygons**Prefixes are also used to name other polygons. tri- 3 triangle quadri- 4 quadrilateral penta- 5 pentagon hexa- 6 hexagon hepta- 7 heptagon octa- 8 octagon nona- 9 nonagon deca- 10 decagon**Q**P R U S T Naming Polygons Terms Consecutive vertices are the two endpoints of any side. A vertex is the point of intersection of two sides. A segment whose endpoints are nonconsecutive vertices is a diagonal. Sides that share a vertex are called consecutive sides.**equiangular**but not equilateral regular, both equilateral and equiangular equilateral but not equiangular Naming Polygons sides An equilateral polygon has all _____ congruent. angles An equiangular polygon has all ______ congruent. equilateral equiangular A regular polygon is both ___________ and ___________. 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 any part of a diagonal lies outside of the figure, then the polygon is _______. If all of the diagonals lie in the interior of the figure, then the polygon is ______. concave convex**Naming Polygons**End of Section 10.1**Vocabulary**Diagonals and Angle Measure What You'll Learn 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? 1 4 2 quadrilateral 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? 1 4 2 quadrilateral 2(180) = 360 2 5 3 pentagon 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? 1 4 2 quadrilateral 2(180) = 360 2 5 3 pentagon 3(180) = 540 3 6 4 hexagon 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? 1 4 2 quadrilateral 2(180) = 360 2 5 3 pentagon 3(180) = 540 3 6 4 hexagon 4(180) = 720 4 7 5 heptagon 5(180) = 900**Diagonals and Angle Measure**1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? 1 4 2 quadrilateral 2(180) = 360 2 5 3 pentagon 3(180) = 540 3 6 4 hexagon 4(180) = 720 4 7 5 heptagon 5(180) = 900 n - 3 n n - 2 n-gon (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. 48° 57° 74° The figure suggests a method for finding the sum of the measures of the exterior anglesof a convex polygon. 72° 55° 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 measures of linear pairs sum of measures of interior angles = – = n•180 – 180(n – 2) = 180n – 180n + 360 sum of measure of exterior angles = 360**Diagonals and Angle Measure**Java Applet**Diagonals and Angle Measure**End of Section 10.2**Vocabulary**Areas of Polygons What You'll Learn 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. 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.**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 smaller and simpler polygon regions, like _______, __________, and ________. rectangles squares triangles adding the The area of the original polygonal region can then be found by __________ _________________________. areas of the smaller polygons**1**2 3 Areas of Polygons AreaTotal = A1 + A2 + A3**Area of Rectangle**1u X 2u = 2u2 Area of Square 3u X 3u = 9u2 3 units 3 units Areas of Polygons Find the area of the polygon in square units. Area of polygon = = 7u2 Area of Rectangle Area of Square**Areas of Polygons**End of Section 10.3**Vocabulary**Areas of Triangles and Trapezoids What You'll Learn You will learn to find the areas of triangles and trapezoids. Nothing new!**h**b 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 _________________. The area of each triangle is half the area of the rectangle, or This result is true of all triangles and is formally stated in Theorem 10-3.**Height**Base Areas of Triangles and Trapezoids Consider the area of this rectangle A(rectangle) = bh**h**b Areas of Triangles and Trapezoids a base of b units, and a corresponding altitude of h units, then**18 mi**6 yd 23 mi Areas of Triangles and Trapezoids Find the area of each triangle: A = 207 mi2 A = 13 yd2**height**base Because the opposite sides of a parallelogram have the same length,the area of a parallelogram is closely related to the area of a ________. Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms. rectangle height The area of a parallelogram is found by multiplying the ____ and the ______. base Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base.**b1**b2 h b2 b1 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. 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.**b1**h b2 Areas of Triangles and Trapezoids bases of b1 and b2 units, and an altitude of h units, then**20 in**18 in 38 in Areas of Triangles and Trapezoids Find the area of the trapezoid: A = 522 in2**Areas of Triangles and Trapezoids**End of Lesson**Vocabulary**Areas of Regular Polygons What You'll Learn 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 polygon is called an ________. apothem congruent In any regular polygon, all apothems are _________.**a**s Areas of Regular Polygons Now, create a triangle by drawing segments from the center to each vertex on either side of the apothem. Now multiply this times the number of triangles that make up the regularpolygon. The figure below shows a center and all vertices of a regular pentagon. The area of a triangle is calculated with the following formula: perpendicular An apothem is drawn from the center, and is _____________ to a side. There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 72 72° 72° 72° 72° 72° What measure does 5s represent? perimeter Rewrite the formula for the area of a pentagon using P for perimeter.**8 ft**5.5 ft Areas of Regular Polygons Find the area of the shaded region in the regular polygon. Area of polygon Area of triangle triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: pentagon The area of the shaded region: 88 ft2 110 ft2 – 22 ft2 =**6.9 m**8 m Areas of Regular Polygons Find the area of the shaded region in the regular polygon. Area of polygon Area of triangle triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: hexagon The area of the shaded region: 110.4 m2 165.6 m2 – 55.2 m2 =**Areas of Regular Polygons**End of Lesson**Vocabulary**Section Title What You'll Learn You will learn to 1)**Vocabulary**Section Title What You'll Learn You will learn to 1)