Day 78

Today’s Agenda Area Rectangles Parallelograms Triangles Trapezoids Kites/Rhombi Circles/Sectors Irregular Figures Regular Polygons

Area – Rectangle On graph paper, draw a rectangle whose sides are on the grid. Count the number of squares inside. How does this total relate to the lengths of the sides? Can we generalize this idea into a formula?

Area – Parallelogram On graph paper, draw a parallelogram with one pair of parallel sides on the grid. Using scissors and tape, see if you can transform the parallelogram into a figure previously studied, without losing area. Can we create a formula with this idea?

Area – Triangle On graph paper, draw two congruent triangles. Using scissors and tape, see if you can transform the triangles into a figure previously studied, without losing area. Can we create a formula with this idea?

Area – Trapezoid On graph paper, draw a trapezoid with its pair of parallel sides on the grid. See if you can transform the trapezoid into a figure(s) previously studied, without losing area. Can we create a formula with this idea?

Area Area is a measurement that describes the amount of space a figure occupies in a plane. Area is a measurement that describes the amount of space a figure occupies in a plane. Area is a two-dimensional measurement. It is measured in square units. Area is a two-dimensional measurement. It is measured in square units. Area Addition Postulate – The area of a region is the sum of the areas of its non- overlapping parts. Area Addition Postulate – The area of a region is the sum of the areas of its non- overlapping parts.

Area Area problems will often refer to the base and height of a figure. Typically (but not always), any side of a figure can act as a base. Area problems will often refer to the base and height of a figure. Typically (but not always), any side of a figure can act as a base. The height must always be perpendicular to the base! The height will typically not be a side of a figure. The height must always be perpendicular to the base! The height will typically not be a side of a figure.

Area of a Rectangle The area of a rectangle is the length of its base times the length of its height. The area of a rectangle is the length of its base times the length of its height. A = bhA = bh BASE HEIGHT

Examples Find the areas of the following rectangles: Find the areas of the following rectangles: ½

The area of a parallelogram is the length of its base times the length of its height. The area of a parallelogram is the length of its base times the length of its height. A = bhA = bh Why? Why? Any parallelogram can be redrawn as a rectangle without losing area. Any parallelogram can be redrawn as a rectangle without losing area. Area of a Parallelogram BASE HEIGHT

Examples Find the areas of the following parallelograms: Find the areas of the following parallelograms:

Area of a Triangle The area of a triangle is one-half of the length of its base times the length of its height. The area of a triangle is one-half of the length of its base times the length of its height. A = ½bhA = ½bh Why? Why? Any triangle can be doubled to make a parallelogram. Any triangle can be doubled to make a parallelogram. BASE HEIGHT

Examples Find the areas of the following triangles: Find the areas of the following triangles:

Area of a Trapezoid Remember for a trapezoid, there are two parallel sides, and they are both bases. Remember for a trapezoid, there are two parallel sides, and they are both bases. The area of a trapezoid is the length of its height times one-half of the sum of the lengths of the bases. The area of a trapezoid is the length of its height times one-half of the sum of the lengths of the bases. A = ½(b 1 + b 2 )hA = ½(b 1 + b 2 )h Why? Why? Red Triangle = ½ b 1 h Red Triangle = ½ b 1 h Blue Triangle = ½ b 2 h Blue Triangle = ½ b 2 h Any trapezoid can be divided into 2 triangles. Any trapezoid can be divided into 2 triangles. HEIGHT BASE 1 BASE 2

Examples Find the areas of the following trapezoids: Find the areas of the following trapezoids:

Area of a Kite/Rhombus The area of a kite is related to its diagonals. The area of a kite is related to its diagonals. Every kite can be divided into two congruent triangles. Every kite can be divided into two congruent triangles. The base of each triangle is one of the diagonals. The height is half of the other one. The base of each triangle is one of the diagonals. The height is half of the other one. A = 2(½ ½d 1 d 2 ) A = 2(½ ½d 1 d 2 ) A = ½d 1 d 2 A = ½d 1 d 2 d1d1d1d1 d2d2d2d2

Area of a Rhombus Remember that a rhombus is a type of kite, so the same formula applies. Remember that a rhombus is a type of kite, so the same formula applies. A = ½d 1 d 2A = ½d 1 d 2 A rhombus is also a parallelogram, so its formula can apply as well. A rhombus is also a parallelogram, so its formula can apply as well. A = bhA = bh

Area of a Circle/Sector Recall the area of a circle: Recall the area of a circle: A = π r 2A = π r 2 Page 782 shows how a circle can be dissected and rearranged to resemble a parallelogram, and how the above formula can be derived. Page 782 shows how a circle can be dissected and rearranged to resemble a parallelogram, and how the above formula can be derived. Recall that the area of a sector is a proportion of the area of the whole circle: Recall that the area of a sector is a proportion of the area of the whole circle: or or

Area of Irregular Figures A composite figure can separated into regions that are basic figures. Add auxiliary lines to divide the figure into smaller sub-figures. – Look to form rectangles, triangles, trapezoids, circles, and sectors. Find the area of each sub-shape. Add the sub-areas together to find the area of the whole figure. – Sometimes you may have to subtract pieces

EXAMPLE  3 = 27 8  3 =  12 = 120 Total Area = = 171 Sq. Units

Example  4 = 8 8  8 = 64 ½  4  8 = 16 Total Area = = 88 Sq. Units

Another Way To Solve…  8 = 96 ½  4  8 = 16 Total Area = – 24 = 88 Sq. Units 4  6 = 24

 Because a regular polygon has unique properties, you only need a little bit of information to find the area.  The basic idea is to dissect the figure as we did before. However, with a regular polygon, we can divide it into congruent isosceles triangles.

 What is the relationship to the number of sides of the polygon and the number of triangles you can draw from the center?  So to find the area of the polygon, we find the area of one of these triangles, and multiply by the number of sides.

 The segment that connects the center of a regular polygon to one of its vertices is called the radius.  This is also a radius of the polygon’s circumscribed circle.

 The segment that connects the center of a regular polygon to the midpoint of one of its sides is the apothem.  The apothem will be perpendicular to that side.  This is also a radius of the polygon’s inscribed circle.

 The apothem also is the height of one of the congruent triangles we drew when dividing the figure up.  So, if we know the height and base of the triangle, we can find its area, and then we multiply by the number of triangles.

 To put it in terms of the polygon, if we know the length of a side ( s ) and the apothem ( a), and the number of sides ( n ), then the area would be: A = (½ as ) n  What would be another way to express s n ? A = ½ ap

 Find the area of the following regular octagon: 12 cm 14.5 cm

 What if we don’t know the apothem?  Is there a way we can calculate it? 22 cm 14.5 cm TRIG!!! Find the area.

Other Triangle Formulas Equilateral Triangle An equilateral triangle with side s can be divided into two triangles. Using the special right triangle ratios, we can represent the height in terms of s. Substituting into the formula A = ½bh… s s s ½ s

Other Triangle Formulas SAS Triangle If we know two sides and an included angle of any triangle, we can use trig to find the area. Drawing the altitude creates a right triangle, of which we know the hypotenuse and angle. Substituting into A = ½bh: C b a h

Other Triangle Formulas Heron’s Formula (SSS) There’s a formula for calculating the area of a triangle if you know the three sides. s in the above formula represents the semi-perimeter, which half of the perimeter b c a

Assignments Homework 46 Workbook, pp. 140, 142 Homework 47 Workbook, pp. 144, 145