11.1 Understanding Area

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1 /6/05. Understanding rea Counting squares is neither the easiest or the best way to find the area of a region. Let s investigate how to find the areas of rectangles and squares Objective: fter studying this section you will be able to understand the concept of area, find the areas of rectangles and squares and use the basic properties of area. ny ideas on how to calculate this in a more timely fashion? We could count them However, that is not the most efficient use of our time. We could multiply the number of columns (base) by the number of rows (height) The Concept of rea We measure lengths of line segments in linear units. It could be in meters, yards, miles, centimeters, etc. The standard units of area or square units such as square meters, square yards, square miles, etc. square mile is the area is the space enclose by a square whose sides are each one mile in length. Definition one square unit one linear unit The area of a closed region is the number of square units of space within the boundary of the region Postulate Postulate The area of a rectangle is equal to the product of the base and the height for that base. rect = bh where b is the length of the base and h is the height In a square, the base and the height are equal. The area of a square is equal to the square of a side. sq = s where s is the length of a side. We can estimate the area of a region by determining the approximate number of square units it would take to fill the region. Postulate Basic Properties of rea Every closed region has an area. estimated area = 4 sq units Postulate If two closed figures are congruent, then their areas are equal. F estimated area = 6 sq units estimated area =? B if BCDEF GHIJKL Then the area of region is congruent to the region of area. E H G C D K L I J

2 /6/05 Postulate If two closed regions intersect only along a common boundary, then the area of their union is equal to the sum of their individual areas. Summary Summarize the postulates you learned about today. = + Homework: worksheet Example Find the area of the rectangle 3 cm 5 cm Example Given that the area of a rectangle is 0 sq dm and the altitude is 5 dm, find the base. (Hint: draw a picture) Example 3 Find the area of the green shaded region

3 /6/05. reas of Parallelograms and Triangles The rea of a Triangle The area of any triangle can be shown to be one half of the area of a parallelogram with the same base and height Objective: fter studying this section you will be able to find the areas of parallelograms and triangles The area of a triangle is equal to one-half the product of a base and the height (or altitude) for that base. = bh where b is the length of the base and h is the altitude. The rea of a Parallelogram We can cut and paste to find the area of a parallelogram. Example Find the area of the triangle height (h) cut base (b) Example 8 mm paste = 3 mm The area of a parallelogram is equal to the product of the base and the height for that base. = bh where b is the length of the base and h is the height Example 3 Find the base of a triangle with altitude 5 and area 60. (Hint: draw a picture) Example 4 Find the area of a parallelogram whose sides are 4 and 6 and whose acute angle is 60 degrees. (Hint: draw a picture).

4 /6/05 Example 5 Find the area of trapezoid (use formulas you learned in this lesson) Summary State in your own words how you can find the area of a parallelogram and a triangle. Homework: worksheet

5 /6/05.3 The rea Of Trapezoid Definition The Median of a Trapezoid The line segment joining the midpoints of the non-parallel sides of a trapezoid is called the median of the trapezoid. Objective: fter studying this section you will be able to find the areas of trapezoids and us the measure of a trapezoid s median to find its area median The rea of a Trapezoid You saw in the last lesson that we can divide a trapezoid into simpler shapes. h b b The measure of the median of a trapezoid equals the average of the measures of the bases. M = b + b ( ) where b is the length of one base and b is the length of the other base. There is a formula though to make finding the area easier. The area of a trapezoid equals one-half the product of the height and the sum of the bases. trap = hb+ b ( ) where b is the length of one base, and b is the length of the other base, and h is the height. If we know the median then we can use a shorter formula to find the area of a trapezoid. The area of a trapezoid is the product of the median and the height. trap = Mh where M is the length of the median and h is the height.

6 /6/05 Example Find the area of the trapezoid cm 7 cm 8 cm Example Find the shorter base of the trapezoid if the area is 5, its altitude is 8, and its longer base is 0. (Hint: draw a picture) Example 3 Find the median of a trapezoid with height and bases 6 and 4. (Hint: draw a picture) Example 4 Find the area of the trapezoid in example 3. Summary State in your own words how you can find the area of a trapezoid. Homework: worksheet

7 /6/05 But Wait!.4 The rea Of a Kite Did you notice BD and C are the diagonals of the kite?! (We just proved the formula for area of a kite no big deal!) Objective: fter studying this section you will be able to find the areas of kites The area of a kite equals half the product of its diagonals. Kite = dd where d is the length of one diagonal, and d is the length of the other diagonal Remember When We Learned Properties of Special Quadrilaterals?. In a kite, the diagonals are perpendicular.. The longer diagonal bisects the shorter diagonal. Just a Note This formula can be applied to any kite, including the special cases of a rhombus and a square d This means the kite can be divided into isosceles triangles with a common base so its area will equal the sum of the areas of the two triangles. d Let s = + Kite ΔBD ΔDBC = BD E + BD EC = ( BD)( E EC) + = ( )( ) ( )( ) ( BD)( C) D E C B Example # Find the area of a kite with diagonals 9 and 4

8 /6/05 Example # Find the area of a rhombus whose perimeter is 0 and whose longer diagonal is 8. Homework Worksheet.4

9 /6/05.5 reas of Regular Polygons Objective: fter studying this section you will be able to find the areas of equilateral triangles and other regular polygons Here are some important observations about apothems and radii: ll apothems of a regular polygon are congruent. Only regular polygons have apothems. n apothem is a radius of a circle inscribed in the polygon. n apothem is the perpendicular bisector of a side. radius of a regular polygon is a radius of a circle circumscribed about the polygon. radius of a regular polygon bisects an angle of the polygon. The rea of an Equilateral Triangle The area of an equilateral triangle equals the product of one-fourth the square of a side and the square root of 3. s eq = 4 3 Where s is the length of a side. The area of a regular polygon equals onehalf the product of the apothem and the perimeter. reg poly = ap Where a is the length of an apothem and p is the perimeter Definition The rea of a Regular Polygon radius of a regular polygon is a segment joining the center to any vertex. Example regular polygon has a perimeter of 40 and an apothem of 5. Find the polygon s area. Definition n apothem of a regular polygon is a segment joining the center to the midpoint of any side. Example n equilateral triangle has a side 0 cm. long. Find the triangle s area

10 /6/05 Example 3 circle with a radius of 6 is inscribed in an equilateral triangle. Find the area of the triangle. Example 4 Find the area of a regular hexagon with sides 8 units long. Summary State in your own words how you can find the area of a polygon and an equilateral triangle. Homework: worksheet

11 /6/05.6 reas of Circles, Sectors, and Segments Objective: fter studying this section you will be able to find the areas of circles, sectors, and segments. The area of a sector of a circle is equal to the area of the circle times the fractional part of the circle determined by the sector s arc. arc measure = π r 360 Where r is the radius and the arc is measured in degrees. Postulate The rea of a Circle The area of a circle is equal to the product of π and the square of the radius. = π r Where r is the radius. Definition The rea of a Segment segment of a circle is a region bounded by a chord of the circle and its corresponding arc. Can you think of a way to calculate the area of a segment? ny ideas? How about now? Definition The rea of a Sector sector of a circle is a region bounded by two radii and the arc of the circle. Example Find the area of a circle whose diameter is 0. Just as the length of an arc is a fractional part of the circumference of a circle, the area of a sector is a fractional part of the area of the circle. Example Find the circumference of a circle whose area is 49 sq units. π

12 /6/05 Example 3 Find the area of a sector with a radius if and a 45 arc. Example 4 The measure of the arc of the segment is 90. The radius of the circle is 0. Find the area of the segment. Summary State in your own words how you can find the area of a segment and a sector. Homework: worksheet

13 /6/05.7 Ratios of reas Objective: fter studying this section you will be able to find ratios of areas by calculating and comparing the areas and applying properties of similar figures. 4 W Similar Figures If two triangles are similar, the ratio of any pair of their corresponding altitudes, medians, or angle bisectors equals the ratio of their corresponding sides. X Y P 6 h Q b 3 h 3 Since = and = b R PQR WXY bh PQR b h = = WXY bh b h = = 4 Computing reas One way to determine the ratio of the areas of two figures is to calculate the quotient of the two areas. Example 9 0 Find the ratio of the area of the parallelogram to the area of the triangle. bh = = bh = = If two figures are similar, then the ratio of their areas equals the square of the ratio of corresponding segments. (Similar Figures ) s = s Where and are areas and s and s are measures of corresponding segments. Example In the diagram, B = 5 and BC =. Find the ratio of the area of triangle BD to triangle CBD. D Given the similar pentagons shown, find the ratio of their areas 9 C B

14 /6/05 Example If BC DEF, find the ratio of the areas of the two triangles. D F 8 E B C Example If the ratios of the areas of two similar parallelograms is 49:, find the ratio of their bases. Example 3 M is the median of triangle BC. Find the ratio BM : CM B M C median of a triangle divides the triangle into two triangles with equal areas. P PQR = PRS Q R S Summary State in your own words how to find the area of a figure using the corresponding segments. Homework: worksheet

15 /6/05.8 Hero s and Brahmagupta s Formulas Example #: Find the area of a triangle with sides 3, 6, and 7. Objective: fter studying this section you will be able to find the areas of figures by using Hero s and Brahmagupta s formula nswer: 4 5 long time ago (about 000 years) a mathematician known as Hero of lexandria developed a formula for finding the area of a triangle. It is as follows: Example #: Find the area of the inscribed quadrilateral with sides, 7, 6, and 9 = s( s a)( s b)( s c) 7 6 Where a, b, and c are the lengths of the sides of the a+ b+ c triangle, and s = semiperimeter =. 9 a b nswer: 30 c In about 68.D., a Hindu mathematician named Brahmagupta found a formula for the area of an inscribed quadrilateral. Homework: Warning: This only applies to quadrilaterals that can be inscribed in circles K Cyclic Quadrilaterals! = ( s a)( s b)( s c)( s d) cyclic quad Where a, b, c, and d are the sides of the quadrilateral and s = semiperimeter = a+ b+ c+ d. a d b c Worksheet.8