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. Area of Shapes Area & Volume Triangles Area = ½ base × perp. height A=½b×h 5cm 6cm A A = ½ ab sin C c b C a B Area = ½ product of 2 sides × sine of angle between them 8cm 60° 12cm Area of Shapes Area & Volume Trapezium One pair of parallel sides b 2 a-b Area 1 → b × h Area 2 → ½ (a – b) × h 1 h a Total area = bh + ½.ah - ½.bh = ½.bh + ½.ah = h/2(a+b) Area of a trapezium = ½ (a +b) × h = h/2(a+b) Area of Shapes Area & Volume Parallelogram Kite b a b a Area = a × b = length × perp. height Area = ½ ab = ½ × length × width Area of Shapes Area & Volume Find the area of the following shapes 1. 5cm 3cm 2. 7cm 10cm 8cm 6cm 12cm Area and Perimeter Area & Volume Circles C A O AB – diameter of circle B E OC – radius of circle DE – chord of circle D Area of a circle A = π r2 Perimeter of a circle Circumference C = π d C=2πr Area & Volume Area and Perimeter Find the area and perimeter of 12cm 8cm 12cm 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. Area & Volume Area and Perimeter 3. The radius of a car wheel is 53cm. How many revolutions does the wheel make in travelling 50m? ? Area & Volume Area and Perimeter 4. Find the shaded area. 8cm 10cm Area & Volume Sectors of a Circle Area and Perimeter O Consider a sector of a circle which subtends an angle of 60° at the centre. 60° r =5cm Find (i) The area of this sector (ii) The arc length AB (iii) The perimeter of the shape OAB A B Area & Volume Sectors of a Circle Area and Perimeter In general θ° Area = θ × π r2 360 r Arc length = θ × 2 π r 360 Sectors, Segments, Chords and Area Area & Volume OA = radius AB = chord O 150° A 7 B The chord AB subtends an angle of 150° at the centre of the circle. The radius of the circle is 7cm. Trigonometry Area & Volume sin x = opposite . hypotenuse Remember SOH CAH TOA cos x = adjacent . hypotenuse tan x = opposite adjacent 2. 1. H 8 x O H 3. 40° 15 x A 30° H x 3 O 40° A O A Volume of Prism Area & Volume Volume of a prism = cross sectional area × length Cylinder V = π r2 h Complete the table below (cylinders) r a) 3 3.2 V 4 5 b) c) h 128 125 Volume of Prism Area & Volume Volume of a prism = cross sectional area × length Trapezoidal V = ½ (a + b)h × l a h l b Example Cross-sectional area = Volume V = 4 5 10 8 Area & Volume Volume of Prism Cone Pyramid h Sphere l h b a a, b = dimensions of base h = perpendicular height V= 1/ 3 × base area × height r r r = radius h = perpendicular height l = slant length r = radius V = 4/3 π r3 V = 1/3 π r2 h Find the volume of (i) Pyramid a = 3, b = 4, h = 2 (ii) Cone r =2, h = 4.5 (iii) Sphere r = 1.5 Area & Volume Volume of Prism Find r. Find h. r h 10 V = 500cm3 V = 250cm3 Find h. Find r. h 6 r V = 300cm3 10 5 V = 210cm3 Surface Area of a Cone, Sphere & Cylinder Area & Volume Sphere Cone Remember… l l h h r l2 = r2 + h2 r r SA = π rl SA =4 π r2 cylinder r SA = Curved surface area + Top + Bottom h 2πr Curved area = = 2 π rh h SA = 2 π rh + 2 π r2 Area & Volume Surface Area of a Cone, Sphere & Cylinder Calculate the Surface Area of a) d) 5 5 3 e) b) l 6 4 6 c) 5.2 r = 1.8 14.8 Area & Volume SOH CAH TOA Use Trig to find angle marked x in the following triangles 3cm x 5cm 3cm 5cm x 3cm 10.4 x 10.4 5cm x 14.6 Using Trig in Area Calculations Area & Volume O 10.4 10.4 14.6 A B Find (i) The angle AOB (ii) The area of sector AOB (iii) Area of triangle AOB (iv) The shaded area Dimensions Area & Volume Distinguishing between length, area & volume Formulae can be used to calculate length e.g. b Volume V = π r2 h r h L2 × L2 = L3 a L+L=L (Volume) (Length) b a Perimeter P = 2a + 2b Area A=a×b L × L = L2 (Area) Area & Volume Further Examples a) 4 l + 4 b + 4 h b) (l + b)2h2 c) (2π)rh d) π(rh2 – 1/3 h3) e) πr (r2 + h2) f) 1/3b2h Dimensions Area & Volume Additional Notes Area & Volume Learning Outcomes: At the end of the topic I will be able to 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. Can Do Revise Further 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. Find the surface area and volume of the cylinder to include inverse calculations