Developing Formulas for Triangles and Quadrilaterals Geometry CP1 (Holt 10-1) K. Santos

Area of a Parallelogram Area = product of its base and height A= bh Base must be perpendicular to the height b h 5cm 3cm 9cm A = 9(3) A = 27 𝑐𝑚 2

Example Find the perimeter of a parallelogram, in which the base is 4ft and the area is 12 𝑓𝑡 2 . Need to find height A = bh 12 = 4h h = 3 ft P = 2b + 2h P = 2(6) + 2(3) P = 18 ft

Area of a Triangle Area = one half of the product of its base and height A= 1 2 bh or A = 𝑏ℎ 2 Base perpendicular to height h h h b b b If b = 4” and h = 6” Then A = 1 2 (4)(6) A = 1 2 (24) A = 12 𝑖𝑛𝑐ℎ𝑒𝑠 2

Example—finding a side The area of a triangle is 24 𝑐𝑚 2 and its height is 3 cm. Find the length of its corresponding base. A = 𝑏ℎ 2 for a triangle 24= 3𝑏 2 48 = 3b b = 16 cm

Area of a Trapezoid Area = (average of the bases)(height) A = 𝑏 1 + 𝑏 2 2 h 𝑏 1 h 𝑏 2 Remember: height is perpendicular to both bases

Example 1--Trapezoid Find the area of the trapezoid. 20 in 25 in 18 in 36 in A = ( 𝑏 1 + 𝑏 2 ) 2 h A = (20 + 36) 2 18 A = (56) 2 18 A = 28(18) A = 504 𝑖𝑛 2

Example 2--Trapezoid A = ( 𝑏 1 + 𝑏 2 ) 2 h find the missing height Find the area of the trapezoid. 11 ft 13 ft 16 ft A = ( 𝑏 1 + 𝑏 2 ) 2 h find the missing height Find the hypotenuse Use Pythagorean theorem 13 2 = 5 2 + ℎ 2 when solved you get …… h = 12 ft A = (11+16) 2 12 A =( 27 2 ) 12 A = 27(6) A = 162 𝑓𝑡 2

Area of a Rhombus The area of a rhombus is half the product of the lengths of its diagonals. A = 𝑑 1 𝑑 2 2 𝑑 2 𝑑 1 Example: Find the area if the diagonals are: 6 in and 8 in A = 𝑑 1 𝑑 2 2 A = 6(8) 2 A = 48 2 A = 24 𝑖𝑛 2

Area of a Kite The area of a kite is half the product of the lengths of its diagonals. 𝑑 1 A = 𝑑 1 𝑑 2 2 𝑑 2 Example 1: Kite with diagonals 9 cm & 8 cm A = 𝑑 1 𝑑 2 2 A = 9(8) 2 A = 72 2 A = 36 𝑐𝑚 2

Example 2--Kite Find the area of the kite. 5” 4” A = 𝑑 1 𝑑 2 2 6” Find diagonals Watch for right triangles 5 2 = 4 2 + 𝑥 2 so x = 3” One diagonal is 2(4) = 8 inches Other diagonal = 3 + 6 = 9 inches A = 8(9) 2 A = 72 2 A = 36 𝑖𝑛 2

Formulas Square: A = bh Rectangle: A = bh Parallelogram: A = bh Trapezoid: A = 𝑏 1 + 𝑏 2 2 h Triangle: A = ½ bh Rhombus: A = 𝑑 1 𝑑 2 2 Kite: A = 𝑑 1 𝑑 2 2

Area Addition Postulate The area of a region is equal to the sum of the areas of its nonoverlapping parts. Best way to find this area is to find the area of rectangle + area of triangle

Example—Partitioning Shapes Find the area of the shape below: 4 9 14 13 16 Find the sum of the areas of the rectangle and the triangle A = bh A = 𝑏ℎ 2 A = 4(14) A = (12)(5) 2 A = 56 A = 30 total area: 56 + 30 = 86 𝑢𝑛𝑖𝑡𝑠 2