Area & Volume

1. Area & Volume • Learning Outcomes • Find the area of square, rectangles, triangles, parallelograms, rhombuses, kites, trapezia and shapes which are composites of these. • Find the circumference and area of a circle, to include inverse calculations. • Find the area and circumference of a sector and segment of a circle. • Find the surface areas and volumes of prisms. • Find the surface area and volume of the cylinder to include inverse calculations • Find the surface area and volumes of the cone and sphere to include inverse calculations. • Find the surface areas and volumes of compound shapes most of which are everyday objects. • Distinguish between formulae for perimeter, area and volume by considering dimensions.

2. 5cm 6cm A c b 8cm B 60° a C 12cm Area & Volume Area of Shapes Triangles Area = ½ base × perp. height A = ½ b × h A = ½ ab sin C Area = ½ product of 2 sides × sine of angle between them

3. Area & Volume b h 1 2 a a - b Area of Shapes Trapezium One pair of parallel sides Total area = bh + ½.ah - ½.bh = ½.bh + ½.ah = h/2(a+b) Area 1→ b × h Area 2 → ½ (a – b) × h Area of a trapezium = ½ (a +b) × h = h/2(a+b)

4. Area & Volume b a b a Area of Shapes Parallelogram Kite Area = a × b = length × perp. height Area = ½ ab = ½ × length × width

5. Area & Volume 5cm 8cm 3cm 7cm 6cm 12cm 10cm Area of Shapes Find the area of the following shapes 1. 2.

6. Area & Volume Area and Perimeter Circles C AB – diameter of circle OC – radius of circle B A O DE – chord of circle E D Area of a circle A = πr2 Perimeter of a circleCircumference C = π d C = 2 πr

7. Area & Volume 12cm 8cm 12cm Area and Perimeter Find the area and perimeter of

8. Area & Volume Area and Perimeter 1. The area of a circle is 51.6cm2. Calculate its radius. 2. The circumference of a circle is 50cm. Calculate its area.

9. Area & Volume Area and Perimeter 3. The radius of a car wheel is 53cm. How many revolutions does the wheel make in travelling 50m? ?

10. Area & Volume 10cm 8cm Area and Perimeter 4. Find the shaded area.

11. Area & Volume O 60° r =5cm A B Sectors of a Circle Area and Perimeter Consider a sector of a circle which subtends an angle of 60° at the centre. • Find • The area of this sector • The arc length AB • The perimeter of the shape OAB

12. Area & Volume Sectors of a Circle Area and Perimeter In general θ° r Arc length =θ× 2 π r 360 Area =θ× π r2 360

13. Area & Volume O 7 150° A B Sectors, Segments, Chords and Area OA = radius AB = chord The chord AB subtends an angle of 150° at the centre of the circle. The radius of the circle is 7cm.

14. Area & Volume H H H A O O O A A Trigonometry sin x = opposite . hypotenuse Remember SOH CAH TOA cos x = adjacent . hypotenuse tan x = opposite adjacent 2. 1. 3. 40° 15 x 8 x x 3 30° 40°

15. Area & Volume Volume of Prism Volume of a prism = cross sectional area × length Cylinder V = πr2 h Complete the table below (cylinders) 4 3 a) 5 128 b) 3.2 125 c)

16. Area & Volume 5 4 a 10 h l 8 b Volume of Prism Volume of a prism = cross sectional area × length Trapezoidal V = ½ (a + b)h × l Example Cross-sectional area = Volume V =

17. Area & Volume h l r h r b a Volume of Prism Cone Pyramid Sphere a, b = dimensions of base h = perpendicular height r = radius h = perpendicular height l = slant length r = radius V = 4/3πr3 V = 1/3× base area × height V = 1/3πr2 h Find the volume of (i) Pyramid (ii) Cone (iii) Sphere a = 3, b = 4, h = 2 r =2, h = 4.5 r = 1.5

18. Area & Volume Find h. h h 10 V = 500cm3 5 10 Volume of Prism Find r. r V = 250cm3 Find h. Find r. 6 r V = 300cm3 V = 210cm3

19. Surface Area of a Cone, Sphere &Cylinder Area & Volume r Remember… l2 = r2 + h2 l l h h r r r 2 πr h h Sphere Cone SA = πrl SA =4 πr2 cylinder SA = Curved surface area + Top + Bottom Curved area = = 2 πrh SA = 2 πrh + 2 π r2

20. Surface Area of a Cone, Sphere &Cylinder Area & Volume 5 Calculate the Surface Area of a) d) 5 3 e) b) l 14.8 6 4 6 c) 5.2 r = 1.8

21. Area & Volume 3cm 3cm x x 10.4 10.4 5cm 5cm 3cm 5cm x x 14.6 SOH CAH TOA Use Trig to find angle marked x in the following triangles

22. Area & Volume O 10.4 10.4 14.6 A B Using Trig in Area Calculations • Find • The angle AOB • The area of sector AOB • Area of triangle AOB • The shaded area

23. Area & Volume r b h a Dimensions Distinguishing between length, area & volume Formulae can be used to calculate length e.g. b Volume V = πr2 h PerimeterP = 2a + 2b a L2× L2 = L3 (Volume) L + L = L (Length) AreaA = a × b L × L = L2 (Area)

24. Area & Volume • 4 l + 4 b + 4 h • (l + b)2h2 • (2π)rh • π(rh2 – 1/3h3) • πr (r2 + h2) • 1/3b2h Dimensions Further Examples

25. Area & Volume Additional Notes

27. Area & Volume Learning Outcomes: At the end of the topic I will be able to Can Revise Do Further • Find the area of square, rectangles, triangles, parallelograms, rhombuses, kites, trapezia and shapes which are composites of these. • Find the circumference and area of a circle, to include inverse calculations. • Find the area and circumference of a sector and segment of a circle. • Find the surface areas and volumes of prisms.        

28.   • Find the surface area and volume of the cylinder to include inverse calculations • Find the surface area and volumes of the cone and sphere to include inverse calculations. • Find the surface areas and volumes of compound shapes most of which are everyday objects. • Distinguish between formulae for perimeter, area and volume by considering dimensions.      