Perimeter, area and volume – of everything! GCSE Maths Lesson 10 Perimeter, area and volume – of everything!

Homework - metric and imperial measures sheet

Perimeter, area and volume

Review LOs Know circle vocabulary and the formula for finding the circumference and the area of a circle Find perimeters, areas and volumes of a variety of 2D and 3D shapes, including triangles, parallelograms, trapeziums, prisms, cones and pyramids, spheres

What do you know already? Can you position the cards in the correct places to show how you would work out the perimeter, area or volume of each shape?

By the end of this lesson you will … Know circle vocabulary and the formula for finding the circumference and the area of a circle Find perimeters, areas and volumes of a variety of 2D and 3D shapes, including triangles, parallelograms, trapeziums, prisms, cones and pyramids, spheres

Perimeters, areas and volumes Give out formulae card Units 1cm 1cm2 1cm3

Perimeter? Area? 4.5m

Perimeter? Area? 7.5m 4.5m

Area? 8cm 12cm

Area?

Perimeter? Area?

Area?

Perimeter 6.3cm 2.1cm 1.2cm 0.8cm Perimeter = 6.3 + 2.1 + 1.2 + 0.8 + 1.2 + 6.3 + 0.8 + 2.1 = 20.8

Area 6.3cm 2.1cm 1.2cm 0.8cm Area = (6.3 x (2.1+0.8)) + (1.2 x 0.8) =(6.3 x 2.9) + (1.2 x 0.8) = 18.27 + 0.96 = 19.23 cm2

Can you … … find the dimensions of a rectangle that has an area of 60 and a perimeter of 34? … find the length and width of a rectangle if its perimeter is 28cm and area is 48cm2? … find the dimensions of a rectangle whose perimeter is 54cm and area is 170cm2?

Surface Area and Volume Cubes, cuboids and other prisms

Prisms A prism is a 3D shape that has a constant cross-section. CUBOID TRIANGULAR PRISM CUBE

A Surface Area Each face is the same – a square. Area A = 5 x 5 = 25cm2 Total Surface Area = 6 x 25 = 150cm2 5cm A 5cm 5cm

Surface Area Surface area is the total area of the outside of a 3D object Area A = 5 x 9 = 45cm2 Area B = 9 x 3 = 27cm2 Area C = 3 x 5 = 15cm2 Total Surface Area = (45 + 27 + 15) x 2 = 174cm2 B C A 5cm 3cm 9cm

C B A Surface Area Area A = 8 x 11 = 88cm2 Area B = 5 x 11 = 55cm2 Area C = 5 x 8 = 40cm2 TOTAL SURFACE AREA = (88 + 55 + 40) x 2 = 183 x 2 = 366cm2 C B 11cm A 5cm 8cm

Surface Area Calculate the Surface Area of the cube and cuboids shown below: SA = 294cm2 SA = 378cm2 15cm 7cm 3cm EXTENSION: 8cm SA = 270cm2 9cm 6cm 12cm 5cm

Practice Calculate the surface area of the following shapes. 5cm 9cm

Volume of Prisms Volume = area of cross-section x depth

Volume of Cuboids A cuboid is a 3-dimensional object made up of a rectangles and squares Volume = height x width x depth Units: cm3, m3, mm3, km3, etc height depth width

Volume of Prisms Volume = width x height x depth V = 4 x 3 x 12 V = 144cm3 12cm 3cm 4cm

Volume of Prisms Volume = Area of cross section x depth V = 7 x 9 x 3 V = 189cm3 7cm 3cm 9cm

Volume of Prisms Volume = Area of cross-section x depth 11cm Area of cross section = (11 x 8) ÷ 2 = 44cm2 9cm 8cm 20cm Volume = 44 x 20 = 880cm3 11cm 2 1 3 7mm 7mm 105m3 343mm3 5cm 80cm3 6m 5m 2cm 7m 8cm 7mm 6m

Worded Problems A length of copper piping is in the shape of a triangular prism. The triangle on it’s end is a right angle, 9mm by 4mm by 10mm. The piece of pipe is 18cm long. What is the surface area of the pipe? Area of Triangles = (9 x 4) ÷ 2 = 18mm2 = 18 x 2 = 36mm2 10mm Front Rectangle = 180 x 10 = 1800mm2 9mm 180mm Back Rectangle = 180 x 9 = 1620mm2 4mm Total Surface Area = 36 + 1800 + 1620 + 720 = 4176mm2 Base Rectangle = 180 x 4 = 720mm2

Take care with metric conversions = 10mm 1cm VOLUME = 1000mm3 VOLUME = 1cm3 10mm 1cm 10mm 1cm x 1000 cm3 mm3 ÷ 1000

Take care with metric conversions = 100cm 1m VOLUME = 100000cm3 VOLUME = 1m3 100cm 1m 100cm 1m x 100000 m3 cm3 ÷ 100000

Measurement Jeopardy.ppt

Review LOs Know circle vocabulary and the formula for finding the circumference and the area of a circle Find perimeters, areas and volumes of a variety of 2D and 3D shapes, including triangles, parallelograms, trapeziums, prisms, cones and pyramids, spheres

How many can you name? How many can you name?

Circumference circumference Circumference = π × diameter diameter

Example 1 Circumference = π × diameter Circumference = π × 4 Find the circumference of this circle circumference Circumference = π × diameter 4cm Circumference = π × 4 = 12·57cm (2 d.p.)

Example 2 Circumference = π × diameter Circumference = π × 16 Find the circumference of this circle circumference Circumference = π × diameter 8cm Circumference = π × 16 = 50·27cm (2 d.p.)

Area Area = π × radius × radius = π × radius2 radius area

Example 1 Area = π × radius × radius Area = π × 7 × 7 Find the area of this circle Area = π × radius × radius 7cm Area = π × 7 × 7 = 153·94cm² (2 d.p.) area

Example 2 Area = π × radius × radius Area = π × 5 × 5 Find the area of this circle Area = π × radius × radius 10cm Area = π × 5 × 5 = 78·54cm² (2 d.p.) area

Find the circumference and area of this circle Question 1 Find the circumference and area of this circle Circumference = π × diameter Circumference = π × 9 = 28·27cm (2 d.p.) 9cm Area = π × radius × radius Area = π × 4·5 × 4·5 = 63·62cm² (2 d.p.)

Find the circumference and area of this circle Question 2 Find the circumference and area of this circle Circumference = π × diameter Circumference = π × 12 = 37·70cm (2 d.p.) 6 cm Area = π × radius × radius Area = π × 6 x 6 = 113·10cm² (2 d.p.)

Surface Area and Volume Cylinders, Cones, Pyramids and Spheres

Real Life Cylinders

Cylinder Volume = r2h What is the volume of this cylinder?

Cones

Cones Pringles A third of the calories If Pringles came in a cone, which was the same height and diameter as the tall tube, it would contain one third of the calories!!! Why?? Pringles A third of the calories

Cones Volume = Slant height (l) Example: find the volume of this cone

Pyramids

Pyramids Creamed Coconut 1/3 of the calories

Pyramids Volume = 1/3 x base area x height Find the volume of this pyramid

Spheres 4/3π of the yumminess of a cube-shaped container!! Find the volume of a sphere whose diameter is 15 cm Volume =

The world's population is presently around 7 billion. The average diameter of the Earth is 6371000m. How much space is this per person? Extension – Only 29% of the earth is land, how much land is there per person?

Surface Area of a Cone, Sphere & Cylinder Area & Volume Sphere Cone l SA = π rl SA =4 π r2 cylinder r h SA = Curved surface area + Top + Bottom h 2 π r Curved area = = 2 π rh SA = 2 π rh + 2 π r2

http://www. mathsbox. org http://www.mathsbox.org.uk/resources/shape/s54/4qa%20(Web) and http://www.mathsbox.org.uk/resources/shape/s54/4qb%20(Web)

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

Complete this table Solid Volume Surface Area D = 10cm L = 15 cm H = 13 cm R = 3.5 cm L = 9 cm H = 60 m L = 50 m W = 40 m D = 23 mm

Mini-Whiteboard Questions Draw a rectangle with: 1. An area of 50cm2 2. An area of 50cm2 and a perimeter of 45cm 3. A perimeter of 40cm 4. A perimeter of 40cm and an area of 75cm2

Example exam questions Topic test – further circumference and area

Review LOs Know circle vocabulary and the formula for finding the circumference and the area of a circle Find perimeters, areas and volumes of a variety of 2D and 3D shapes, including triangles, parallelograms, trapeziums, prisms, cones and pyramids, spheres

Review LOs Know circle vocabulary and the formula for finding the circumference and the area of a circle Find perimeters, areas and volumes of a variety of 2D and 3D shapes, including triangles, parallelograms, trapeziums, prisms, cones and pyramids, spheres

Homework

Perimeter, area and volume - Homework

What is the value of x? 6

The area of this shape is 20cm2. What is the length, b ? 4cm b cm 5

What is the perimeter of this shape? 2.5cm 2.5cm 2 cm 7

What is the volume of this shape? 4

2 The perimeter of this shape is 7cm. What is the missing length x? x cm

What is the area of this shape? 12

What is the surface area of this shape? 18

The area of this shape is 46cm2. What is the value of y? 23

The area of this square is 100cm2. What is the length of each side? 10

1 The perimeter of this rectangle is 18cm What is the value of y? y cm

3

The area of this shape is 130cm2. What is the length, b ? 13 10cm b cm

11 The perimeter of this rectangle is 28cm What is the value of y? 3cm

8

6cm 4cm 6cm 9

What is the perimeter of this shape? 14

15

What is the area of this shape? 19 9.5cm

What is the perimeter of this shape? 16

What is the area of this shape? 17 8.5cm

3cm What is the area of this shape? 4cm 20 Clue

21 The perimeter of this shape is 71cm. What is the missing length x? x cm

What is the perimeter of this shape? 22 7cm 4cm

The area of this shape is 20cm2. What is the perimeter? 24

Also for homework SYWAGC – circles

Also for homework? Topic test circles

Two-way tables and frequency trees.

How could we display the information below in an easier format How could we display the information below in an easier format? A school chess club has 70 members of which 40 are boys. Students play in competitions on a regular basis. Last month, 13 girls and 11 boys played in competitions. 2 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

A two way table might be helpful…. Played in competition last month Did not play in competition last month Boys 11 29 40 Girls 13 17 30 24 46 70 What other methods might we use to display the information? 3 of 4 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

11 40 Boys 29 70 13 Girls 30 17 Use a frequency tree Played in competition 11 40 Boys Didn’t play in competition 29 70 Played in competition 13 Girls 30 17 Didn’t play in competition 4 of 4 Copyright © 2015 AQA and its licensors. All rights reserved. AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

x of x Version 3.0 Copyright © AQA and its licensors. All rights reserved.

x of x Version 3.0 Copyright © AQA and its licensors. All rights reserved.

In Year 6 at a local primary school there are 120 students In Year 6 at a local primary school there are 120 students. The ratio of boys to girls is 9:6. The girls were twice as likely to own a mobile phone as they were to not own a mobile phone. The ratio of boys who own a mobile phone to those who don’t own a mobile phone is 5:3 2 of 3 AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

45 72 Boys 27 120 32 Girls 48 16 Use a frequency tree Owns a mobile phone 45 72 Boys Doesn’t own a mobile phone 27 120 Owns a mobile phone 32 Girls 48 Doesn’t own a mobile phone 16 3 of 3 Copyright © 2015 AQA and its licensors. All rights reserved. AQA Education (AQA) is a registered charity (number 1073334) and a company limited by guarantee registered in England and Wales (number 3644723). Our registered address is AQA, Devas Street, Manchester M15 6EX.

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Last bit of probability for now - tree diagrams

TREE DIAGRAMS First coin Second coin H H T H T T

H T

H T

Imagine choosing a ball from this bag and then replacing it Imagine choosing a ball from this bag and then replacing it. If you did this three times, what's the probability that you would pick at least one green ball? What’s the best method to use to answer this question? What if you didn't replace the ball each time?

Replacing it - if you did this three times, what's the probability that you would pick at least one green ball?

Not replacing it - if you did this three times, what's the probability that you would pick at least one green ball?

Two cards are drawn from a pack with replacement 13 52 39 A spade Not a spade 1st card 2nd card

Two cards are drawn from a pack without replacement 13 52 39 A spade Not a spade 1st card 2nd card

Exam question 11

Looking back at targets Do you understand that the sum of the probabilities of all possible mutually exclusive outcomes is 1? Do you understand the difference between theoretical and experimental probability? Can you calculate probability using a tree diagram?

Probability revision quiz HIGHER In teams of 3 work together to answer all the questions

Finding probabilities 1 (words) 5 points 10 points 15 points P(roll an odd number on a dice) is P(it will rain tomorrow) is P(baby born is a girl) is P(being younger tomorrow) is P(win lottery) is P(sun rising tomorrow) is

Finding probabilities 2 (fractions) 5 points (dice) 10 points (cards) 15 points P(red prime)= P(black)= P(4)= P(even)= P(red picture card)= P(less than 5) =

Finding probabilities 3 (OR rule) 5 points (dice) 10 points (cards) 15 points P(prime or square)= P(6 or 7)= P(4 or 5)= P(2, 3 or 4)= P(red 3 or black queen)= P(2 or 3) =

Finding probabilities 4 (2 events) 5 points (2 dice) 10 points 15 points P(2 numbers the same)= P(5 or 6)= P( 7 )= P(13)= P(less than 4) = P(even) = You have 1 minute to list all the outcomes of 2 dice before the questions come up.

Finding probabilities 5 (biased dice) 1 2 3 4 5 6 0.2 0.1 0.05 x 0.15 5 points 10 points 15 point P(not 4)= P(6)= P(2)= P(1 or 3)= P(not 2)= P(odd) =

Finding probabilities 6 (tree diagrams) Copy down this information: The probability it rains on a Monday is 0.3 The probability it rains on Tuesday is 0.25 5 points 20 points 30 points P( it rains on both days) Draw a tree diagrams to show all outcomes P( it does not rain on Monday)= P(It does not rain on Tuesday)=

Bonus question (40 points) Based on the following tree diagram what is the probability I pick 2 different colours?

Relative frequency in a graph

PLENARY Who wants to be a millionaire? millionaire_probability.ppt

Need to do recurring decimals

Still to do Use pi in an answer