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SLIDE 1
The area of a triangle is its base
Example
The area of this triangle is
Square centimetres is written , so we write this
SLIDE 1B
The area of a triangle is its base
Example
The area of this triangle is
Square inches is written , so we write this
SLIDE 2
In each case, find the area. Give the area of the last in
SLIDE 3
? ?
? ?
SLIDE 2B
In each case, find the area. Give the area of the last in
SLIDE 3B
? ?
? ?
SLIDE 4
Find the area of the shape
SLIDE 5
Area = area of first triangle + area of square + area of second triangle
Area
SLIDE 4B
Find the area of the shape.
SLIDE 5B
Area = area of first triangle + area of square + area of second triangle
Area
SLIDE 6
Find the area of the shape
SLIDE 7
The shape must be first divided into a rectangle and a triangle.
The base of the triangle is 3.5 cm and its height is 8 cm.
Area = area of ? + area of ?
Area
SLIDE 6B
Find the area of the shape
SLIDE 7B
The shape must be first divided into a rectangle and a triangle.
The base of the triangle is 3.5 in and its height is 8 in.
Area = area of ? + area of ?
Area
SLIDE 8
Find the area of the shape
SLIDE 9
The shape must be first divided into a rectangle and two triangles
The height of ? is
The width of ? is
Area = area of ? + area of ? + area of ?
Area
SLIDE 8B
Find the area of the shape
SLIDE 9B
The shape must be first divided into a rectangle and two triangles
The height of ? is
The width of ? is
Area = area of ? + area of ? + area of ?
Area
SLIDE 10
Find the area of the shape
SLIDE 11
We could solve this problem either by dividing the shape into one rectangle and two triangles, or, as in the second diagram, subtracting triangle ? from rectangle ?.
The idea of subtracting an area to find an area is important in practical problems, so we use that method here.
Area = area of ? area of ?
Area
SLIDE 10B
Find the area of the shape
SLIDE 11B
We could solve this problem either by dividing the shape into one rectangle and two triangles, or, as in the second diagram, subtracting triangle ? from rectangle ?.
The idea of subtracting an area to find an area is important in practical problems, so we use that method here.
Area = area of ? area of ?
Area
SLIDE 12
A regular hexagon is inscribed within a circle of diameter 6 cm
You are also told that the distance from one side to another is 5.60 cm to 2 dp
Find the area of this regular hexagon
SLIDE 13
The hexagon can be divided into six equilateral triangles. The side of each of these is a radius of the circle, so of base 3 cm and height half 5.60 = 2.80 cm
Area
The result is given to 2 dp because we were given the distance from one side to another of the hexagon as 5.60 cm to 2 dp.
SLIDE 12B
A regular hexagon is inscribed within a circle of diameter 6 in.
You are also told that the distance from one side to another is 5.60 in to 2 dp
Find the area of this regular hexagon
SLIDE 13B
The hexagon can be divided into six equilateral triangles. The side of each of these is a radius of the circle, so of base 3 in and height half 5.60 = 2.80 in
Area
The result is given to 2 dp because we were given the distance from one side to another of the hexagon as 5.60 in to 2 dp.
SLIDE 14
This shape is symmetrical
The dimensions of the height of this shape are the same as the dimensions shown for the width
Find the area of the shape
SLIDE 15
The shape is made of 1 rectangle of dimensions 9 cm by 9 cm and four triangles of base 9 cm and height 3 cm
Area
SLIDE 14B
This shape is symmetrical.
The dimensions of the height of this shape are the same as the dimensions shown for the width
Find the area of the shape
SLIDE 15B
The shape is made of 1 rectangle of dimensions 9 in by 9 in and four triangles of base 9 in and height 3 in
Area
SLIDE 16
Find the area of the triangle
SLIDE 17
The base of the triangle is (x coordinates)
The height of the triangle is (y coordinates)
Area
SLIDE 18
Onto the graph paper plot the triangle whose vertices are
Find the area of this triangle
SLIDE 19
SLIDE 20
Onto the graph paper plot the shape whose vertices are successively
Find the area of this shape
SLIDE 21
The shape comprises one rectangle and two triangles
SLIDE 22
Divide this shape into one square and four triangles. There are two pairs of triangles with equal areas. Match these up, and find the area of the shape without calculation as the sum of the areas of two squares and a rectangle.
SLIDE 23
SLIDE 24
We cannot find the exact area of an irregular shape.
We can find an approximate area.
We draw a grid onto the shape, and the rule is – a square is counted as in if it is more than half inside the shape. You judge this by eye, and if the square is half in then it is in.
Then you count the squares. If the size of each of the squares above is then the approximate area of the shape because there are 25 squares that are in.
SLIDE 25
Each square is
Approximate the area of this shape
SLIDE 26
41 squares are in
Approximate area is
SLIDE 24B
We cannot find the exact area of an irregular shape.
We can find an approximate area.
We draw a grid onto the shape, and the rule is – a square is counted as in if it is more than half inside the shape. You judge this by eye, and if the square is half in then it is in.
Then you count the squares. If the size of each of the squares above is then the approximate area of the shape because there are 25 squares that are in.
SLIDE 25B
Each square is
Approximate the area of this shape.
SLIDE 26B
41 squares are in.
Approximate area is
SLIDE 27
Each square represents the size of a square tile. A gentleman wishes to tile his patio in the above shape. Each tile will cost £24 and tiles that are cut are not wasted. Find the cost (assuming no breaks, accidents or wastage) of tiling the patio.
SLIDE 28
The shape is made up of a square of side 6 by 6 tiles and two pairs of triangles of the same size. The combined size of one of these pairs is 2 tiles and the combined size of the other pair is 4 tiles.
SLIDE 27b
Each square represents the size of a square tile. A gentleman wishes to tile his patio in the above shape. Each tile will cost $24 and tiles that are cut are not wasted. Find the cost (assuming no breaks, accidents or wastage) of tiling the patio.
SLIDE 28b
The shape is made up of a square of side 6 by 6 tiles and two pairs of triangles of the same size. The combined size of one of these pairs is 2 tiles and the combined size of the other pair is 4 tiles.
SLIDE 29
A floor is to be tiled with four kinds of tile. White tiles cost £20, blue tiles cost £25, pink tiles cost £30 and red tiles cost £40. What is the total cost of tiling the floor?
SLIDE 30
A floor is to be tiled with four kinds of tile. White tiles cost £20, blue tiles cost £25, pink tiles cost £30 and red tiles cost £40. What is the total cost of tiling the floor?
Solution
The rectangle has tiles in total
There are pink tiles and 19 blue tiles
There are 10 red tiles (counting two halves as one)
So, there are white tiles
SLIDE 29B
A floor is to be tiled with four kinds of tile. White tiles cost $20, blue tiles cost $25, pink tiles cost $30 and red tiles cost $40. What is the total cost of tiling the floor?
SLIDE 30B
A floor is to be tiled with four kinds of tile. White tiles cost $20, blue tiles cost $25, pink tiles cost $30 and red tiles cost $40. What is the total cost of tiling the floor?
Solution
The rectangle has tiles in total
There are pink tiles and 19 blue tiles
There are 10 red tiles (counting two halves as one)
So, there are white tiles
SLIDE 31
A larger area of open ground is to be covered with a lawn
Turf comes in roles
Work out how many roles of turf are required
SLIDE 32
The area of the lawn is
One roll of turf is
SLIDE 31B
A larger area of open ground is to be covered with a lawn
Turf comes in roles
Work out how many roles of turf are required
SLIDE 32B
The area of the lawn is
One roll of turf is
SLIDE 33
The lawn is to be surrounded by a gravel path. The width of the path is to be 4 m. It requires 1 bag of gravel to fill . Work out the number of bags of gravel required for the path.
SLIDE 34
The lawn is to be surrounded by a gravel path. The width of the path is to be 4 m. It requires 1 bag of gravel to fill . Work out the number of bags of gravel required for the path.
Solution
We could divide the patio up into a series of rectangles, but it is much quicker to calculate the area of the patio as the bigger area less the area of the lawn. From the previous question
By adding 8 m to the various dimensions of the lawn, we obtain the dimensions of the larger shape as shown.
SLIDE 33B
The lawn is to be surrounded by a gravel path. The width of the path is to be 4 yd. It requires 1 bag of gravel to fill . Work out the number of bags of gravel required for the path.
SLIDE 34B
Solution
We could divide the patio up into a series of rectangles, but it is much quicker to calculate the area of the patio as the bigger area less the area of the lawn. From the previous question
By adding 8 yd to the various dimensions of the lawn, we obtain the dimensions of the larger shape as shown.
SLIDE 35
An ornamental pond is to be surrounded by an alabaster pavement. All the dimensions are given in metres. Find the area of the alabaster surround. If the alabaster is to be 40 cm thick, what is the total volume of alabaster required in cubic metres?
SLIDE 36
An ornamental pond is to be surrounded by an alabaster pavement. All the dimensions are given in metres. Find the area of the alabaster surround. If the alabaster is to be 40 cm thick, what is the total volume of alabaster required in cubic metres?
Solution
SLIDE 35B
An ornamental pond is to be surrounded by an alabaster pavement. All the dimensions are given in yards. Find the area of the alabaster surround.
SLIDE 36B
An ornamental pond is to be surrounded by an alabaster pavement. All the dimensions are given in yards. Find the area of the alabaster surround.
Solution
SLIDE 37
Books are placed onto two shelves
Both shelves are 0.84 metres long
The books are all 30 mm wide
What is the maximum number of books that can be placed on the two shelves?
SLIDE 38
Books are placed onto two shelves. Both shelves are 0.84 metres long. The books are all 30 mm wide. What is the maximum number of books that can be placed on the two shelves?
Solution
SLIDE 37B
Books are placed onto two shelves
Both shelves are 0.85 yd long
The books are all 1.2 in wide
What is the maximum number of books that can be placed on the two shelves?
SLIDE 38B
Solution
SLIDE 39
Parallelogram
A parallelogram is a figure made of two pairs of parallel lines
Because the lines are parallel, they are always the same distance apart, so the parallel sides are equal in length
The area of a parallelogram is the base
Question
Are squares and rectangles parallelograms?
SLIDE 40
Squares and rectangles are special cases of parallelograms
We can represent this idea in a Venn diagram
This diagram represents using concentric circles the following two statements
(A) All squares are rectangles
(B) All rectangles are parallelograms
Draw Venn diagrams to represent the following statements
? All parallelograms are quadrilaterals
? All regular hexagons are hexagons
? All scalene triangles, isosceles triangles, equilateral triangles and right-angled triangles are triangles. All equilateral triangles are isosceles triangles. No scalene triangle is either an equilateral or isosceles triangle. Some right-angled triangles are scalene. Some right-angled triangles are isosceles.
SLIDE 41
? All parallelograms are quadrilaterals
? All regular hexagons are hexagons
? All scalene triangles, isosceles triangles, equilateral triangles and right-angled triangles are triangles. All equilateral triangles are isosceles triangles. No scalene triangle is either an equilateral or isosceles triangle. Some right-angled triangles are scalene. Some right-angled triangles are isosceles.
Notes. The size of the shape does not indicate a number of items. All these collections are infinite. Any shape can be used. For example, in the last illustration we use a rectangle to represent “all triangles”.
SLIDE 42
? ?
Find the area of these two parallelograms
SLIDE 43
? ?
SLIDE 42B
? ?
Find the area of these two parallelograms
SLIDE 43B
? ?
? ?
SLIDE 44
In the above arrangement the overall length is 27 m and the overall height is 12 m. The base of each triangle is 3.5 m. Find the area of the green part, the area of the blue part and ratio of the green to the blue.
SLIDE 45
In the above arrangement the overall length is 27 m and the overall height is 12 m. The base of each triangle is 3.5 m. Find the area of the green part, the area of the blue part and ratio of the green to the blue.
Although the blue parts are also parallelograms, it is easiest to see this design as made of blue triangles and green parallelograms. The height of 12 m makes the height of the triangle and parallelogram 3 m. The base of the triangle is 3.5. Subtracting twice this length from the overall length and dividing by 4, gives the base of the parallelogram as 5 m.
There are 16 triangles and 16 parallelograms.
There are many alternative methods to find the answer. For example, each blue diamond is made of two triangles.
SLIDE 44B
In the above arrangement the overall length is 27 yd and the overall height is 12 yd. The base of each triangle is 3.5 yd. Find the area of the green part, the area of the blue part and ratio of the green to the blue.
SLIDE 45B
Solution
Although the blue parts are also parallelograms, it is easiest to see this design as made of blue triangles and green parallelograms. The height of 12 yd makes the height of the triangle and parallelogram 3 yd. The base of the triangle is 3.5. Subtracting twice this length from the overall length and dividing by 4, gives the base of the parallelogram as 5 yd.
There are 16 triangles and 16 parallelograms.
There are many alternative methods to find the answer. For example, each blue diamond is made of two triangles.
SLIDE 46
A trapezium is a quadrilateral where there are a pair of parallel lines. If these two parallel lines are equal in length then the trapezium is a parallelogram.
The area of a trapezium is the average of the lengths of the two parallel lines times its height.
? “All parallelograms are trapezia, but not all trapezia are parallelograms.” Is this statement true or false?
? In the above diagram , and . Find the area of the trapezium.
SLIDE 47
Trapezium
? “All parallelograms are trapezia, but not all trapezia are parallelograms.” This statement is true.
? , and
SLIDE 46B
A trapezium is a quadrilateral where there are a pair of parallel lines. If these two parallel lines are equal in length then the trapezium is a parallelogram.
The area of a trapezium is the average of the lengths of the two parallel lines times its height.
? “All parallelograms are trapezia, but not all trapezia are parallelograms.” Is this statement true or false?
? In the above diagram , and . Find the area of the trapezium.
SLIDE 47B
Trapezium
? “All parallelograms are trapezia, but not all trapezia are parallelograms.” This statement is true.
? , and
SLIDE 48
Represent in a single Venn diagram the relationships between quadrilaterals, rectangles, parallelograms, and trapezia.
Express these relationships as a series of statements involving the word “all”.
SLIDE 49
All rectangles are parallelograms
All parallelograms are trapezia
All trapezia are quadrilaterals
SLIDE 50
Find the ratio of the area of the triangle to the area of the trapezium
SLIDE 49
SLIDE 52
A kite is a quadrilateral with two pairs of equal sides
A rhombus is a quadrilateral where all the sides are equal
Another name for a rhombus, not used in geometry, is a diamond
TASK
The area of both the kite and the rhombus is the product* of the lengths of their diagonals. By dividing both shapes into four triangles, show that this is true.
* Recall that the term product means multiply. Here we multiply the two diagonals and divided by 2.
SLIDE 53
Kite
We draw in diagonals and divide the kite into two pairs of triangles equal in area. The length of one diagonal is and the length of the other is a.
Rearrange the figure so that it makes two rectangles. The area of the two rectangles together (also a rectangle) is
In other words, it is half the product of the lengths of the two diagonals.
The same argument works for the rhombus, but is easier, because the rhombus is made of four equal triangles.
SLIDE 54
Find the ratio of the area of the shaded part to the area of the whole.
SLIDE 55
The area of the whole (a rhombus) is half the product of its diagonals.
Area
The area of the kite (shown in yellow) is
The difference between the rhombus and the kite is and the area of the shaded triangle is half this, 12
SLIDE 56
In the above the upward arrow means, “All squares are rectangles”.
Make a complete diagram for all the shapes.
SLIDE 57
SLIDE 58
Example
All squares are rhombuses. All squares are rectangles.
Some rhombuses are squares. Not all rhombuses are squares.
Task
Convert the following to Venn diagrams
SLIDE 59
SLIDE 60
By making a parallelogram into a rectangle with the same area, demonstrate the formula for the area of the parallelogram
SLIDE 61
SLIDE 62
By dividing a parallelogram into two, prove that the area of a triangle is
SLIDE 63
By dividing a parallelogram into two parts, we obtain two identical triangles of equal area. The area of the parallelogram is the product of its base and height. Therefore,
SLIDE 64
The word altitude is another word for the height of a triangle
Question
A triangle has a base of 8 cm and an altitude of 5 cm. Calculate its area.
SLIDE 65
A triangle has a base of 8 cm and an altitude of 5 cm. Calculate its area.
Solution
SLIDE 64B
The word altitude is another word for the height of a triangle
Question
A triangle has a base of 8 in and an altitude of 5 in. Calculate its area.
SLIDE 65B
A triangle has a base of 8 in and an altitude of 5 in. Calculate its area.
Solution
SLIDE 66
? The area of a triangle is 40 cm2. Its base is 8 cm. What is its altitude?
? The parallel sides of a trapezium are 12 cm and 8 cm. Its area is 120 cm2. What is its height?
SLIDE 67
? The area of a triangle is 40 cm2. Its base is 8 cm. What is its altitude?
? The parallel sides of a trapezium are 12 cm and 8 cm. Its area is 120 cm2. What is its height?
Area is half the product of the average of the two parallel sides and the height
SLIDE 66B
? The area of a triangle is 40 in2. Its base is 8 in. What is its altitude?
? The parallel sides of a trapezium are 12 in and 8 in. Its area is 120 in2. What is its height?
SLIDE 67B
? The area of a triangle is 40 in2. Its base is 8 in. What is its altitude?
? The parallel sides of a trapezium are 12 in and 8 in. Its area is 120 in2. What is its height?
Area is half the product of the average of the two parallel sides and the height
SLIDE 68
? A white isosceles triangle of altitude 10 cm and base 7 cm is mounted within the frame of a blue rectangle of height 15 cm and base 10 cm. Find the area of the blue rectangle that remains visible.
? A rectangular lawn of dimensions 12 m by 8 m is surrounded by a pavement of width 2.5 m. What is the area of the pavement?
SLIDE 69
? A white isosceles triangle of altitude 10 cm and base 7 cm is mounted within the frame of a blue rectangle of height 15 cm and base 10 cm. Find the area of the blue rectangle that remains visible.
Area of rectangle
Area of triangle
Area of blue visible
? A rectangular lawn of dimensions 12 m by 8 m is surrounded by a pavement of width 2.5 m. What is the area of the pavement?
Area of lawn
Dimensions of larger surround are
and
Area of surround
Area of pavement
SLIDE 68B
? A white isosceles triangle of altitude 10 in and base 7 in is mounted within the frame of a blue rectangle of height 15 in and base 10 in. Find the area of the blue rectangle that remains visible.
? A rectangular lawn of dimensions 12 yd by 8 yd is surrounded by a pavement of width 2.5 yd. What is the area of the pavement?
SLIDE 69B
? A white isosceles triangle of altitude 10 in and base 7 in is mounted within the frame of a blue rectangle of height 15 in and base 10 in. Find the area of the blue rectangle that remains visible.
Area of rectangle
Area of triangle
Area of blue visible
? A rectangular lawn of dimensions 12 yd by 8 yd is surrounded by a pavement of width 2.5 yd. What is the area of the pavement?
Area of lawn
Dimensions of larger surround are
and
Area of surround
Area of pavement
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