Composite Shapes Circular Parts with Circular Parts © T Madas

© T Madas

π π A semicircle has a radius of 9 cm. Calculate its area Calculate its perimeter 127 cm2 π A = x r 2 c π A = x 92 c 9 cm A ≈ 254 cm2 © T Madas

π π A semicircle has a radius of 9 cm. Calculate its area Calculate its perimeter 28.25 cm π C = 2 x x r c π C = 2 x x 9 c 9 cm C ≈ 56.5 cm P = 28.25 + 18 = 46.25 cm © T Madas

© T Madas

π π A quarter-circle has a radius of 16 cm. Calculate its area Calculate its perimeter π 201 cm2 A = x r 2 c π A = x 182 c 16 cm A ≈ 804 cm2 © T Madas

π π A quarter-circle has a radius of 16 cm. Calculate its area Calculate its perimeter 25.1 cm π C = 2 x x r c π C = 2 x x 16 c 16 cm C ≈ 100.53 cm P = 25.1 + 16 + 16 = 57.1 cm © T Madas

© T Madas

Calculate the perimeter and area of the following composite shape. 1 2 x π P = 10 + 8 + 10 + x 2 x 4 4 cm x π P = 28 + 4 10 cm P ≈ 40.6 cm 8 cm x π C = 2 x r © T Madas

π π Calculate the perimeter and area of the following composite shape. 1 2 x π P = 10 + 8 + 10 + x 2 x 4 4 cm x π P = 28 + 4 A1 cm 10 cm P ≈ 40.6 1 2 π A = 10 x 8 + x x 4 2 8 cm x π A = 80 + 8 x π C = 2 x r A ≈ 105 cm2 π A = x r 2 © T Madas

© T Madas

Calculate the perimeter & area of the grey region below. 6 cm © T Madas

π Calculate the perimeter & area of the grey region below. The perimeter of the grey area is equal to … … the circumference of a circle of radius … … 3 cm 6 cm x π P = 2 x 3 P ≈ 18.8 cm x π C = 2 x r π A = x r 2 © T Madas

π π Calculate the perimeter & area of the grey region below. The area of the grey area is equal to … … the area of a square with side 6 cm … … less … … the area of a circle of radius 3 cm 6 cm π A = 6 x 6 – x 3 2 x π x π C = 2 x r A = 36 – 9 π A = x r 2 A ≈ 7.73 cm2 © T Madas

© T Madas

π π π π [ ] x Calculate the perimeter & area of the following shape The perimeter is equal to … … the circumference of a semi-circle of radius 20 cm … … plus … … the circumference of a circle of radius 10 cm … 10 cm 40 cm x π [ ] x 1 2 x π P = 2 x 20 + 2 x 10 π π P = 20 + 20 x π C = 2 x r π P = 40 π A = x r 2 P ≈ 125.7 cm © T Madas

π π π π π [ ] x Calculate the perimeter & area of the following shape The total area is equal to … … the area of a semi-circle of radius 20 cm … … plus … … the area of a circle of radius 10 cm … 10 cm 40 cm π [ ] x 1 2 + π A = x 20 2 x 10 2 π π A = 200 + 100 x π C = 2 x r π A = 300 π A = x r 2 A ≈ 942 cm2 © T Madas

© T Madas

Calculate to 2 decimal places: the perimeter of the pond. The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m. Calculate to 2 decimal places: the perimeter of the pond. the area of the pond 4 m 4 m © T Madas

The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m. Calculate to 2 decimal places: the perimeter of the pond. the area of the pond C = 2 x π x r c 4 4 m 6.28 ? C = 2 x π x 4 c 4 C ≈ 25.13 m Each curved edge is ¼ of the circumference of a full circle. 4 6.28 ? 25.13 ÷ 4 ≈ 6.28 m 4 P = 4 x 4 + 2 x 6.28 ≈ 28.57 m © T Madas

The figure below shows a pond made up of two squares and two identical quarter circles with a radius of 4 m. Calculate to 2 decimal places: the perimeter of the pond. the area of the pond A = π x r 2 c 16 m2 A = π x 4 2 c 12.57 m2 A = π x 16 c 4 m 4 m A ≈ 50.27 m2 12.57 m2 16 m2 50.27 ÷ 4 ≈ 12.57 m2 Area of a Quarter Circle P = 2 x 16 + 2 x 12.57 ≈ 57.14 m2 © T Madas

© T Madas

π π [ ]x 4 Calculate the perimeter & area of the following shape: The perimeter is equal to … … the circumference of … … 8 semi-circles of radius 4 m … … or … … 4 circles of radius 4 m … 4 m 16 m x π P = [ ]x 4 2 x 4 π P = 32 P ≈ 100.5 m x π C = 2 x r π A = x r 2 © T Madas

π π [ ]x 4 Calculate the perimeter & area of the following shape: The total area is equal to … … the area of a square with side length of 16 m … … plus … … the area of 4 circles of radius 4 m … 4 m 16 m [ ]x 4 π A = 16 x 16 + x 4 2 x π A = 256 + 64 x π C = 2 x r A ≈ 457 m2 π A = x r 2 © T Madas

© T Madas

π π π π [ ] x Calculate the perimeter & area of the following shape: The perimeter is equal to … … the circumference of a semi-circle of radius 14 cm … … plus … … the circumference of a circle of radius 7 cm … 7 cm 28 cm x π [ ] x 1 2 x π P = 2 x 14 + 2 x 7 π π x π P = 14 + 14 C = 2 x r π P = 28 π A = x r 2 P ≈ 88.0 cm © T Madas

π π π π π [ ] x Calculate the perimeter & area of the following shape: The total area is equal to … … the area of a semi-circle of radius 14 cm … … less… … the area of a circle of radius 7 cm … 7 cm 28 cm π [ ] x 1 2 – π A = x 14 2 x 7 2 π π x π A = 98 – 49 C = 2 x r π A = 49 π A = x r 2 A ≈ 154 cm2 © T Madas

Harder Problems © T Madas

Break my Heart... © T Madas

Find the area of the heart in terms of a Area of square: 2 semi-circles = circle Note that the circle’s radius is: a a © T Madas

© T Madas

© T Madas

Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them. © T Madas

Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them. 1 solution © T Madas

Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them. Ac = π r 2 c Ac = π x 12 c Ac = π the total area is 1 the area of this composite consists of a circle of unit radius plus a 1 by 2 rectangle © T Madas

Calculate in terms of π the area of the composite shape drawn below which consists of three semicircles of unit radius and the area enclosed by them. Ac = π r 2 c Ac = π x 12 c Ac = π the total area is π + 2 1 the area of this composite consists of a circle of unit radius plus a 1 by 2 rectangle © T Madas

© T Madas

The grey area... © T Madas

Find the area enclosed by the 4 circles in terms of a Area of the square: a 4 quarter-circles = circle © T Madas

© T Madas

...flower petals... © T Madas

Find the exact area of the orange “petal” Billy wants a hint ... a © T Madas

© T Madas

Find the exact area of the orange “petal” one of the blue regions: area of the square less the area of the quarter circle both blue regions a The area of the “petal” is given by the area of the square less the area of the two blue regions: © T Madas

© T Madas

Vase equals Square © T Madas

? Vase equals Square Look at this vase shaped object It consists of 6 identical arcs Each arc is a quarter circle If the quarter circles to which these arcs correspond have radius a, find the area of this object a a ? © T Madas

? Vase equals Square Look at this vase shaped object It consists of 6 identical arcs Each arc is a quarter circle If the quarter circles to which these arcs correspond have radius a, find the area of this object a a ? 1st hint 2nd hint © T Madas

Vase equals Square a a 1st hint 2nd hint © T Madas

Vase equals Square 4a 2 No complex calculations needed ! 2a The area of this object is equal to the area of the square on the right. No complex calculations needed ! Vase equals Square a a 4a 2 2a 1st hint 2nd hint © T Madas

another approach ... © T Madas

© T Madas

Yin Yang © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections a © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections a © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections A r e a working with the green section: a These semicircles both have a radius of © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections A r e a working with the green section: The area of the green section of the Yin Yang is equal to the area of a semicircle a These semicircles both have a radius of © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections A r e a working with the green section: a © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections P e r i m e t e r working with the green section; the required perimeter is given by: the circumference of a circle of diameter plus the circumference of a semicircle of diameter a © T Madas

In the following Yin – Yang symbol calculate the area and perimeter of each of its two identical sections P e r i m e t e r a © T Madas

© T Madas

? Spiral Galaxies Comets Yin Yang Marbles © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. a © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. a © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. The blue sections are congruent Area of the blue sections a This semicircle has a radius of © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. Area of the blue sections a This semicircle has a radius of This semicircle has a radius of © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. Area of the blue sections a © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. Area of the orange section This is best found by subtracting the areas of the two blue sections we just found from the whole circle a © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. perimeter of a blue section a © T Madas

Look at the shape below, consisting of three sections. Calculate the area and perimeter of these sections. perimeter of a blue section perimeter of the orange section a © T Madas

The generalisations of the Yin Yang shape: The circle in every case : is divided by curved lines of equal lengths the resulting regions have equal perimeters the resulting regions have equal areas © T Madas

© T Madas

... another grey area... © T Madas

Find the grey area enclosed by the 3 circles in terms of a The grey area is equal to the area of an equilateral triangle of side 2a less a semicircle of radius a a Area of Triangle: 2a 60° 2a © T Madas

Find the grey area enclosed by the 3 circles in terms of a The grey area is equal to the area of an equilateral triangle of side 2a less a semicircle of radius a a Area of Triangle: Area of semicircle: The grey area: © T Madas

© T Madas

Broomsticks and Elastic Bands Broomsticks and Elastic Bands © T Madas

How long is the elastic band in terms of a ? Three cylindrical broomsticks each of radius a are held together by an elastic band. How long is the elastic band in terms of a ? Solution all the distances between the centres of the circles are 2a , so we have an equilateral triangle at the centre. 60° a © T Madas

How long is the elastic band in terms of a ? Three cylindrical broomsticks each of radius a are held together by an elastic band. How long is the elastic band in terms of a ? Solution 120° Draw radii as shown towards the elastic band. The radii must be at right angles at the points of contact with the elastic band. (tangent – radius) We can now work another useful angle 60° a © T Madas

How long is the elastic band in terms of a ? Three cylindrical broomsticks each of radius a are held together by an elastic band. How long is the elastic band in terms of a ? Solution 120° We can now calculate some lengths. Each straight piece (not in contact with the circles) has length 2a Each arc corresponds to one third of a circle Finally do all the adding 2a 60° a © T Madas

How long is the elastic band in terms of a ? Three cylindrical broomsticks each of radius a are held together by an elastic band. How long is the elastic band in terms of a ? Solution 120° 2a 60° a © T Madas

© T Madas

Annulus Problem 1 © T Madas

3 4 5 Three concentric circles have radii of 3, 4 and 5 units, as shown opposite. What percentage of the largest circle is shaded? Annulus: So: © T Madas

© T Madas

Train Sets © T Madas

D Marcus has two circular railway lines, one with radius of 1.5 metres and the other with radius of 2 metres. He runs an engine clockwise round each track at the same speed from the start line of the diagram. Where would the engine on the outer track be, out of A, B, C or D when the engine on the inner track has made 11 complete circuits? 2 1.5 C A B © T Madas

D C A B 2 A circuit on the inner track: A circuit on the outer track: 11 complete circuits on the inner track: Both engines travel at the same speed, so the engine on the outer track must also cover a distance of 33π, with each circuit in the outer track being 4π 2 1.5 C A B The engine on the outer track will be at point B when the engine on the inner track has completed 11 circuits. © T Madas

© T Madas