Section 8.2 Perimeter and Area of Polygons The perimeter of a polygon is the sum of the lengths of all sides of the polygon. Table 8.1 and 8.2 p. 363 Ex 1,2 p. 364 11/24/2018 Section 7.2 Nack

Theorem 8.2.1 Heron’s Formula for the Area of a Triangle Semiperimeter s = ½ (a + b + c) Heron’s Formula: A = s (s - a) (s - b) (s - c) Example 3: Find the area of a triangle with sides 4, 13, 15. s = ½ (4 + 13 + 15) = 16 A = 16 (16 - 4) (16 - 13) (16 - 15) = 24 sq. units. 11/24/2018 Section 7.2 Nack

Theorem 8. 2. 2: Brahmagupta’s Formula for the area of a cyclic Theorem 8.2.2: Brahmagupta’s Formula for the area of a cyclic* quadrilateral Semiperimeter = ½(a + b + c + d) Area = A =  (s - a) (s - b) (s - c) (s – d) *cyclic quadrilateral can be inscribed in a circle so that all 4 vertices lie on the circle. 11/24/2018 Section 7.2 Nack

Area of a Trapezoid Theorem 8.2.3: The area A of a trapezoid whose bases have lengths b1 and b2 and whose altitude has length h is given by: A = ½ h (b1 + b2 ) = ½ (b1 + b2)h The average of the bases times the height Proof p. 366 Example 4 p. 367 11/24/2018 Section 7.2 Nack

Quadrilaterals with Perpendicular Diagonals Theorem 8.2.4: The area of any quadrilateral with perpendicular diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ). Corollary 8.2.5: The area A of a rhombus whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ) Corollary 8.26: The area A of a kite whose diagonals of lengths d1 and d2 is given by A = ½ ( d1d2 ) Proof: Draw lines parallel to the diagonals to create a rectangle. The area of the rectangle A = ( d1d2 ). Since the rectangle is twice the size of the kite, the area of the kite A = ½ ( d1d2 ) Ex. 6 p. 369 11/24/2018 Section 7.2 Nack

Areas of Similar Polygons Theorem 8.2.7: The ratio of the areas of two similar triangles equals the square of the ratio of the lengths of any two corresponding sides: Proof p. 369 Note: This theorem can be extended to any pair of similar polygons (squares, quadrilaterals, etc.) Ex. 7 p. 370 11/24/2018 Section 7.2 Nack