Chapters 1.18 and.18 Areas, Perimeters and Volumes In this chapter, we will learn about: From Text Book 1: 1. Perimeter of a flat shape: 1.A Perimeter of a square 1.B Perimeter of a rectangle 1.C Perimeter of a composite shape. Area of a flat shape:.a Area of a square.b Area of a rectangle.c Area of a composite shape 3. Different units From Text Book : 4. Area of a triangle: 4.A Area of a right triangle 4.B Area of any triangle 4.C Area of composite shapes 5. Area of a parallelogram and of a trapezium 5.A Area of a parallelogram 5.B Area of a trapezium 6. Cuboids 6.A Surface area of a cuboid 6.B Volume of a cuboid 7. The circle 7.A Definition and the elements of a circle 7.B Circumference of a circle 7.C Area of a circle 1
Important Note: In this chapter we will learn many new formulas. These formulas are most easily remembered by noting that they all follow the same idea of length, which will produce formulas for perimeters, the idea of surface, which will produce the formulas for areas and the idea of space occupied, which will produce the formula for volume. If we understand these 3 main ideas, memorization of these formulas is almost unnecessary. Text Book 1: 1. Perimeter of a flat shape: Definition 1: For a given flat polygon, the perimeter of the polygon if the sum of all the sides of a polygon. 1.A. Perimeter of a square: Therefore, for a square of side a as in Figure 1 : Figure 1: A square of side a The Perimeter of the square is : (1) P 4 s a 1B. Perimeter of a rectangle: For a rectangle of length l and width w as shown in Figure : Figure : A rectangle of length l and width w
The Perimeter of the rectangle is : () P l w l w 1.C Perimeter of a composite shape: r For a composite shape, calculate the perimeter of the shape by adding all of the sides of the shape.. Area of a flat shape: Area of a flat shape represents how many square units that flat shape occupies. Definition : The area of a polygon is the number of square units which that polygon occupies..a. Area of a square: Therefore, for a square with side a units as shown in Figure 1, (3) As a units..b Area of a rectangle: For a rectangle with length l (units) and width w (units), the area it occupies is: (4) Ar l w units. For example, a rectangle with length 3 (units) and width (units) occupies a total area of 3 6 units. Example: From Exercise set 18.1 (Text Book 1) do problems and 5..C. Area of a composite shape: We calculate the area of a composite shape either by breaking it down into a sum of rectangles, or using difference of rectangles. Worked Example (Page 156 Text Book 1): 3
We can do this using either sum or difference of rectangles. Example: From Exercise set 18. (Text Book 1) do problems 1, and 5. 3. Different units: If different units are involved, use the following transformations to bring all length to the same units, before calculating areas and / or perimeters: 1 m 100 cm (5) 1 cm 10 mm 1 m = 1000 mm Example: From Exercise Set 18.1 (Book 1) do problems 1, and 6. Text Book : 4. Area of a triangle: 4.A. Area of a right triangle: Consider a right triangle as shown in Figure 3: Figure 3: A right triangle of height h and base length b 4
Then its area is given by the formula: (6) A T base height Reason: See why by representing this triangle as half of a certain rectangle. 4.B. Area of any triangle: Consider a generic triangle as shown in Figure 4: Figure 4: A generic acute triangle of height h and base length b Then its area is given by the same formula: (7) A T base height Reason: See why by splitting this triangle in two right triangles (or use a different reasoning). Note that formula (7) works for an obtuse triangle as shown in Figure 5 as well. Figure 5: A generic obtuse triangle of height h and base length b Why? 5
Therefore, we consider formula (7) as true for the area of ANY triangle of base length b and height length h. Example: From Exercise Set 18.. A (Text Book ) do problems 1, a) and 3 and from Exercise Set 18.. B (Text Book ) do problem 1. 5. Area of a parallelogram and of a trapezium: 5.A. Area of a parallelogram: Figure 6: A parallelogram of height h and base length b The area of a parallelogram of height h and base length b as shown in Figure 6 above is: (8) AP b h To see why, see that the area of this parallelogram is the area of a rectangle with length b and width h. 5.B. Area of a trapezium: 6
Figure 7: A trapezium with length of parallel sides a and b and with height h The area of a trapezium as shown in Figure 7 above is: (8) A Trapezium a b h To see why, split the trapezium into a rectangle and two right triangles. Note: Memorize the area formulas: (3), (4), (6), (7) and (8) as we will use these often. Example: From Exercise Set 18.3 A in Text Book do problems 1 and 3 and from Exercise Set 18.3. B do problem. 6.Cuboids: A cuboid is a 3 dimensional box as shown in Figure 8. Figure 8: A cuboid of length l, width w and height h. 6.A. Surface area of a cuboid Its surface area is the sum of the areas of its faces (that is the total area covered by its faces, if the cuboid were to be expanded over a D flat surface). We can see that the surface area of a general cuboid of length l, width w and height h is: (9) S l w l h w h l w l h w h C Example: Consider the cuboid shown below: units 7
Figure 9: A specific cuboid Calculate its surface area. 6.B Volume of a cuboid: The volume of a cuboid is the number of cube units (that is the number of cubes of a side of length 1 unit) which it contains. The volume of a cuboid of length l, width w and height h is given by the formula: (10) V l w h 3 units Example: From Exercise Set 18.4 do problems 1 and 5. 6. The Circle 7.A. Definition and elements of a circle: As we have seen in Chapter 10, we have: Definition: Given a fixed point in plane O, and a positive number (the radius of the circle R), the circle with center O and radius R is the set of points in plane P which are at the fixed distance r from O, so for which: OP R. A generic circle is shown in Figure 10 below: 8
Figure 10: A circle of center O and radius R. We already see certain elements of the circle: the center O, the radius R (which is also the segment OP), the diameter D (which is the segment through the center of the circle which has as endpoints two points on the circle), and the circumference C (the circumference can be thought of as the perimeter of the circle). Other elements of the circle are shown in Figure 11 below. 9
Figure 11: Main elements of a circle. Describe in your words: the chord of a circle, the segment of a circle, the arc of a circle, the sector of a circle, and the tangent to a circle. Example: Do Exercise 18.5.A from Book. 7.B. The circumference of a circle: Activity 1: Let us measure the circumference of a circle. We need: 7. One cylinder (cup) or a bicycle wheel; 8. One piece of string; 9. A ruler; 10. Paper and pencil. Steps: Step 1: Measure the diameter of the circle; Step : Wrap the string tightly around the edge of the cylinder and measure the circumference of the circle. Step 3: Record your results; Step 4: Divide the circumference to the diameter. What number did you obtain? This should produce the following formula for the circumference of a circle: (11) C D R Example: From Exercise Set 18.5. B (Book ) do exercises 1 b, d, and c. 7.C. The area of a circle: In order to find the area of a circle, we split it in many sectors. More sectors we consider, better the approximation of our final formula will be. Consider Figure 1 below: 10
Figure 1: Deriving the formula for the area of a circle. Note that the number of sectors is the same in the top and bottom figures. This suggests the following formula for the area of a circle of radius R: (1) Ac R Example: From Exercise Set 18.5 C do exercises a, e and f. Homework: What is left from Exercises above and from Text Book 1: Exercise Set 18.1: problems 3 and 4, and Exercise Set 18.: problems 3 and 4. From Text Book : Exercise Set 18. : problem, Exercise Set 18.3.A : problem, Exercise Set 18.3. B: problems 1 and 3, Exercise Set 18.4: problems and 6. 11