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

Area & Volume 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 h b a 2 1 a - b Area 2 → ½ (a – b) × h Area of a trapezium = ½ (a +b) × h = h / 2 (a+b)

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

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

Area & Volume Area and Perimeter Circles O C B A E D AB – diameter of circle OC – radius of circle DE – chord of circle Area of a circle A = π r 2 Perimeter of a circle Circumference C = π d C = 2 π r

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

Area & Volume Area and Perimeter 1. The area of a circle is 51.6cm 2. 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 Consider a sector of a circle which subtends an angle of 60° at the centre. Find (i) The area of this sector (ii) The arc length AB (iii) The perimeter of the shape OAB 60° r =5cmO B A

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

Area & Volume Sectors, Segments, Chords and Area O BA 7 150° 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.

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

Area & Volume Volume of Prism Volume of a prism = cross sectional area × length Cylinder V = π r 2 h rhV Complete the table below (cylinders) a) b) c)

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

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

Area & Volume Volume of Prism Find r. r 6 V = 300cm 3 Find r. r V = 250cm 3 10 h V = 500cm 3 Find h. h 5 10 Find h. V = 210cm 3

Area & Volume Surface Area of a Cone, Sphere & Cylinder r h l Cone SA = π rl h r l l 2 = r 2 + h 2 Remember… r h cylinder SA = 2 π rh + 2 π r 2 SA = Curved surface area + Top + Bottom h 2 π r Curved area = = 2 π rh r Sphere SA =4 π r 2

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

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

Area & Volume Using Trig in Area Calculations O BA Find (i)The angle AOB (ii)The area of sector AOB (iii) Area of triangle AOB (iv)The shaded area

Area & Volume Dimensions Distinguishing between length, area & volume r h Formulae can be used to calculate length e.g. Volume V = π r 2 h L 2 × L 2 = L 3 (Volume) b a Perimeter P = 2a + 2b L + L = L (Length) b a Area A = a × b L × L = L 2 (Area)

Area & Volume Dimensions Further Examples a)4 l + 4 b + 4 h b)(l + b) 2 h 2 c)(2π)rh d)π(rh 2 – 1 / 3 h 3 ) e) πr (r 2 + h 2 ) f) 1 / 3 b 2 h

Area & Volume Additional Notes

Area & Volume  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. Learning Outcomes: At the end of the topic I will be able to Can Revise Do Further        

 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.        