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 What You'll Learn Naming Polygons What You'll Learn You will learn to name polygons according to the number of _____ and ______. sides angles Vocabulary 1) regular polygon 2) convex 3) concave

A polygon is named by the number of its _____ or ______. sides angles Naming Polygons A polygon is a _____________ in a plane formed by segments, called sides. closed figure A polygon is named by the number of its _____ or ______. sides angles A triangle is a polygon with three sides. The prefix ___ means three. tri

Prefixes are also used to name other polygons. Prefix Number of Sides 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

Terms Naming Polygons A vertex is the point of intersection of two sides. Consecutive vertices are the two endpoints of any side. U T S Q R P A segment whose endpoints are nonconsecutive vertices is a diagonal. Sides that share a vertex are called consecutive sides.

An equilateral polygon has all _____ congruent. sides Naming Polygons An equilateral polygon has all _____ congruent. sides An equiangular polygon has all ______ congruent. angles A regular polygon is both ___________ and ___________. equilateral equiangular 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?

A polygon can also be classified as convex or concave. 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 polygon is ______. If any part of a diagonal lies outside of the figure, then the polygon is _______. concave convex

Diagonals and Angle Measure What You'll Learn You will learn to find measures of interior and exterior angles of polygons. Vocabulary Nothing New!

Diagonals and Angle Measure Number of Diagonals from One Vertex 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 Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral 4 1 2 2(180) = 360

Diagonals and Angle Measure Number of Diagonals from One Vertex 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 Number of Diagonals from One Vertex 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 Number of Diagonals from One Vertex 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 Number of Diagonals from One Vertex 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 Theorem 10-1 If a convex polygon has n sides, then the sum of the 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. 48° 57° 74° The figure suggests a method for finding the sum of the measures of the exterior angles of 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 Theorem 10-2 In any convex polygon, the sum of the measures of the exterior angles, (one at each vertex), is 360. Java Applet

Vocabulary What You'll Learn Areas of Polygons What You'll Learn You will learn to calculate and estimate the areas of polygons. Vocabulary 1) polygonal region 2) composite figure 3) irregular figure

Any polygon and its interior are called a ______________. Areas of Polygons Any polygon and its interior are called a ______________. polygonal region In lesson 1-6, you found the areas of rectangles. Postulate 10-1 Area Postulate For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the 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.

The figures above are examples of ________________. composite figures Areas of Polygons The figures above are examples of ________________. composite figures 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 ________. squares rectangles triangles The area of the original polygonal region can then be found by __________ _________________________. adding the areas of the smaller polygons

Area Addition Postulate Areas of Polygons Postulate 10-3 Area Addition Postulate The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon. 1 2 3 AreaTotal = A1 + A2 + A3

Find the area of the polygon in square units. Areas of Polygons Find the area of the polygon in square units. Area of Square 3u X 3u = 9u2 Area of Rectangle 1u X 2u = 2u2 Area of polygon = = 7u2 3 units Area of Rectangle Area of Square

Areas of Triangles and Trapeziods What You'll Learn You will learn to find the areas of triangles and trapezoids. Vocabulary Nothing new!

Areas of Triangles and Trapezoids Look at the rectangle below. Its area is bh square units. The diagonal divides the rectangle into two _________________. congruent triangles 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. b h

Areas of Triangles and Trapezoids Height Consider the area of this rectangle A(rectangle) = bh Base

Areas of Triangles and Trapezoids Theorem 10-3 Area of a Triangle If a triangle has an area of A square units, a base of b units, and a corresponding altitude of h units, then h b

Areas of Triangles and Trapezoids Find the area of each triangle: 6 yd 18 mi 23 mi A = 13 yd2 A = 207 mi2

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 base The area of a parallelogram is found by multiplying the ____ and the ______. base height 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 b1 h 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.

Areas of Triangles and Trapezoids Theorem 10-4 Area of a Trapezoid If a trapezoid has an area of A square units, bases of b1 and b2 units, and an altitude of h units, then b1 b2 h

Areas of Triangles and Trapezoids Find the area of the trapezoid: 18 in 20 in 38 in A = 522 in2

Areas of Regular Polygons What You'll Learn You will learn to find the areas of regular polygons. Vocabulary 1) center 2) apothem

Areas of Regular Polygons Every regular polygon has a ______, a point in the interior that is equidistant from all the vertices. center A segment drawn from the center that is perpendicular to a side of the regular polygon is called an ________. apothem In any regular polygon, all apothems are _________. congruent

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 regular polygon. The area of a triangle is calculated with the following formula: The figure below shows a center and all vertices of a regular pentagon. An apothem is drawn from the center, and is _____________ to a side. perpendicular There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 72 72° 72° 72° 72° s a 72° What measure does 5s represent? perimeter Rewrite the formula for the area of a pentagon using P for perimeter.

Areas of Regular Polygons Theorem 10-5 Area of a Regular Polygon If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then P

Areas of Regular Polygons Find the area of the shaded region in the regular polygon. 5.5 ft 8 ft Area of polygon Area of triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle pentagon 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 Area of triangle 6.9 m 8 m To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle hexagon The area of the shaded region: 165.6 m2 – 55.2 m2 = 110.4 m2

Areas of Regular Polygons Geometer's Sketchpad

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