area & volume

Area & VolumeLearning 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. heightA = 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

21

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

BA

E

D

AB – diameter of circle

OC – radius of circle

DE – chord of circle

Area of a circle A = π r2

Perimeter of a circleCircumference C = π d C = 2 π r

Area & Volume Area and Perimeter

Find the area and perimeter of

12cm

12cm

8cm

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 CircleArea 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 =5cm

O

BA

Area & Volume Sectors of a CircleArea and Perimeter

In general

θ° r

Area = θ × π r2

360

Arc length = θ × 2 π r 360

Find(i) The area of the sector OAB (ii) The area of the triangle

OAB

(i) The area of the minor segment (shaded)

Area & Volume Sectors, Segments, Chords and Area

O

BA

7150°

OA = radiusAB = 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

3x

40°

3.

O

A

H

Area & Volume Volume of Prism

Volume of a prism = cross sectional area × length

Cylinder

V = π r2 h

r h V

3

3.2

4

5 128

125

Complete the table below (cylinders)

a)

b)

c)

lh

a

b

Area & Volume Volume of Prism

Volume of a prism = cross sectional area × length

Trapezoidal

V = ½ (a + b)h × l

104 5

8

Cross-sectional area =

Volume V =

Example

Area & Volume Volume of Prism

ConePyramid Sphere

h

ba

r

hl

r

a, b = dimensions of base h = perpendicular height

r = radiush = perpendicular heightl = slant length V = 4/3 π r3

V = 1/3 π r2 hV = 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 = 300cm3

Find r.

r

V = 250cm3

10

h

V = 500cm3

Find h.

h

510

Find h.

V = 210cm3

Area & VolumeSurface Area of a

Cone, Sphere & Cylinder

r

hl

Cone

SA = π rl

h

r

l l2 = r2 + h2

Remember…

r

h

cylinder

SA = 2 π rh + 2 π r2

SA = Curved surface area + Top + Bottom

h

2 π rCurved area = = 2 π rh

r

Sphere

SA =4 π r2

Area & VolumeSurface Area of a

Cone, Sphere & Cylinder

Calculate the Surface Area of

3

5

b)

6

l6

a)

c)

5.2

r = 1.8

5

d)

e)14.8

4

Area & Volume SOH CAH TOA

Use Trig to find angle marked x in the following triangles

5cm3cm

x

3cm

5cm

x

3cm

5cm

x

10.410.4

14.6

x

Area & Volume Using Trig in Area Calculations

O

BA

10.4

14.6

10.4

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.

VolumeV = π r2 h

L2 × L2 = L3

(Volume)

b

a

PerimeterP = 2a + 2b

L + L = L

(Length)

b

a

Area A = a × b

L × L = L2

(Area)

Area & Volume Dimensions

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

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.