Areas of some geometric gures

Transcription

1 reas of some geometric gures The area of a given geometric gure is the number of unit squares, (i.e. squares with length and width equal to unit), that may be tted into the region. The simplest is a rectangle like the one in gure (i) below. It has length 6 cm and width cm. Its area, in square centimeters, is the number of squares with length and width equal to cm each, that can be tted into the rectangle. cm 6 cm Figure (i) Figure (ii) s shown in Figure (ii), they are 0, therefore its area is 0 square centimeters, abbreviated to 0 sq. cm. It should be easy to see that the area of a rectangle with length l units and width w units is lw sq. units. Of course areas do not have to be whole numbers. The area of the trapezium below is 7 trapezium The trapezium sliced into squares We sliced it into unit squares as shown in the gure to the right. There are full squares plus half squares for a total of 7 square centimeters. The net simple gure to consider is a parallelogram like the one shown in Figure (iii) below. It has vertices at (0; 0), (; 0), (; ) and (; ). 0 D 6 Figure (iii)

2 To gure out the number of unit squares that t into it, cut out triangle, shown in gure (iv) and paste it as shown in gure (v). The result is a rectangle with corners at (; 0), (; 0), (; ) and (; ) shown in Figure (vi). total of = 0 unit squares t into the rectangle, therefore the parallelogram has area 0 square units Figure (iv) Figure (v) Figure (vi) We now introduce some terms used in computing areas of parallelograms: Referring to the parallelogram D in Figure (iii), The side is called a base for the parallelogram. In this case it has length units. The length of the line segment is called the height of the parallelogram. Thus a height of a parallelogram is the distance between two parallel sides of the parallelogram. The parallelogram in this case has geight. Note that the area of this parallelogram is = 0 square unit which happens to be the product of its length and its height. In general, rea of a parallelogram = (Length of parallelogram) (Its Height) The third gure we consider is a triangle. Figure (a) shows a triangle. We pick one of its three sides and call it the base of the triangle. For convenience, we have picked. We then drop a perpendicular from the verte opposite the base, which is, to the base itself. This is the line segment D in Figure (b). The length of D is called the height of the triangle. The triangle may be converted into a parallelogram, shown in Figure (c), as follows: Imagine that is a mirror. Now draw a re ection of triangle in the mirror. The result is the parallelogram. D D Figure (a) Figure (b) The base of the parallelogram is, and its height is the length of D, therefore its area is equal to (Length of base ) (Length of D) Note that the area of the parallelogram is twice the area of the triangle. Therefore rea of triangle = (Length of the ase of ) (Height of ) It turns out that this result holds for any given triangle. In other words: The rea of a Triangle = (Length of its ase) (Its Height)

3 ample The area of triangle, in the gure below, with a base of length cm and height 9 cm is ( 9) = 0: sq. cm. 9 The height may not be given. In such a case, use the given information to calculate it. ample In triangle below, =, b = cm and c = cm. Take as the base of the triangle. It has length cm. (i) (ii) In gure (ii), we have added the height D of the triangle. Since Length of D = sin Length of it follows that the height of the triangle is (Length of ) (sin ) = () (0:7). Therefore rea of Triangle = () () (0:7) = 0: sq. cm. We rounded o the answer to decimal places. rea of a SS triangle In general, to calculate the area of a given SS triangle, follow the steps described above. Say you are given the angle and the two sides b and c as shown below. Take as the base. The length of the base is c and the height of the triangle is b sin. Therefore the area of the triangle is bc sin b Given angle c ample To calculate the area of triangle with a = ft., = 67 and b = 9 ft. and round o the answer to dec. pl. Solution We are given two sides with lengths ft. and 9 ft. We are also given the angle between these two sides. It is = 67. Therefore the area of the triangle is rea = 9 sin 67 = 9:7

4 rea of a S triangle If you are given two angles and a side, (i.e. you are given a S triangle), calculate one of the unknown sided. You will then have two sides and an angle between them and you may calculate the area of the triangle as described above. ample To calculate the area of triangle with =, = 7 and b = cm. Solution If we calculate c then we will have two sides and an angle between them. (The two sides are c and b; the angle between them is.) Since = (0 7 ) = 6, c is given by It follows that c = rea of a SSS triangle c sin 6 = sin 7 : sin 6 sin 7. The area of the triangle is bc sin which translates into sin 6 sin 7 sin = : sq. cm. to decimal place. If you are given the lengths of the three sides of a triangle and no angle, you may calculate its area using Heron s formula. Say the sides have lengths a, b, and c. Start by determining the number s = (a + b + c). Then according to Heron s theorem, the area of the triangle is p s (s a) (s b) (s c) ample To calculate the area of the triangle with a = 0 cm, b = cm and c = cm. 0 Solution: In this triangle, s = ( ) = :. Therefore s a = :, s b = : and s c = 6:. y Heron s formula, the area of the triangle is p (:) (:) (:) (6:) = 9: square centimeters, (to decimal place)

5 ercise 6 alculate the area of each triangle (i) (ii) 6 (iii) (iv)