11.4 Areas of Irregular Figures

Objectives Find areas of irregular figures. Find areas of irregular figures on the coordinate plane.

Areas of Irregular Figures An irregular figure is a figure that cannot be classified into the specific shapes that we have studied. Irregular figures are also called composite figures because the region can be separated into smaller regions. D F E Auxiliary lines are drawn in quadrilateral ABCD. DE, and DF separate the figure into ADE, CDF, and rectangle BEDF

Postulate 11.2 The area of a region is the sum of all of its nonoverlapping parts.

Example 1: Find the area of the figure. The figure can be separated into a rectangle with dimensions 6 units by 19 units, a semicircle with a radius of 3 units, and an equilateral triangle with sides each measuring 6 units. º Use the 30º-60º-90º relationships to find that the height of the triangle is 3Ö3.

Example 1: = 112.5 units² = lw – ½ bh + ½ (pi)(r)² Area Formulas Area of irregular figure= Area of rectangle – area of triangle + area of semicircle = lw – ½ bh + ½ (pi)(r)² Area Formulas = 19(6) – ½(6)(3Ö3) + ½(pi)(3²) Substitution = 114 – 9Ö3 + ½(9)(pi) Simplify Use a calculator = 112.5 units²

Areas of Irregular Figures on a Coordinate Plane V (6, 0) To find the area of an irregular polygon on the coordinate plane, separate the polygon into known figures.

Example 2: Find the area of the shaded region. Find the difference between x-coordinates to find the length of the base of the triangle and the lengths of the bases of the trapezoid. Find the difference between the y-coordinates to find the heights of the triangle and trapezoid. S (-3, 7) U (6, 7) T (4, 11) R (-5, 0) V (6, 0)

Example 2: = 88.2 units² Area of STU + area of trapezoid RSUV Area of RSTUV= Area of STU + area of trapezoid RSUV = ½bh + ½h(b1+b2) = ½(8.1)(4.5) + ½(7)(9+11) = 88.2 units² Area formulas Substitution Simplify

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