2 What Is Area ? Area is the amount of space inside a shape: Area Area 1cmArea is measured in square centimetres.1cm2A square centimetre is a square measuring one centimetre in each direction.It is written as :
3 Estimating The Area. B A C D Look at the four shapes below and use your judgement to order them from smallest to largest area:ABCD
4 B A C D To decide the order of areas consider the four shapes again: To measure the area we must determine how many square centimetres are in each shape:Each shape is covered by 36 squares measuring a centimetre by a centimetre .We can now see that all the areas are equal at 36cm2 each.
5 Area Of A Rectangle.Look again at one of the shapes whose area we estimated:CLengthBreadthWhat was the length of the rectangle ?9cmHow many rows of 9 squares can the breadth hold ?4We can now see that the area of the rectangle is given by 9 x 4.The formula for the area of a rectangle is:A = LBfor short.Area = Length x Breadthor
6 We can now calculate the area of each rectangle very quickly: (1)(2)A= L x BA = 12 x 3 =36cm2(3)A= L x BA = 6 x 6 =36cm2A= L x BA= L x B(4)A = 18 x 2 =36cm2A = 9 x 4 =36cm2
7 Example 1Calculate the area of the rectangle below:(2)3m5m7cm4cm(1)SolutionThis area is in square metres:1mA = LBSolutionA = LBL = 7B = 4L = 3B = 5A = 7 x 4A = 3 x 5A = 28cm2A = 15m2
8 Example 3.Solution.8cm2cm5cm3cmSplit the shape up into two rectangles:A1Calculate the area of A1 and A2.A22A1A2356Calculate the area of the shape above:Area = A1 + A2Area = ( 2 x 5) + (6 x 3)Area =Area = 28cm2
9 What Goes In The Box ? Find the area of the shapes below : (1) 8cm 6cm (2)48cm2(3)17cm8cm12cm5cm11.34m2141cm2
10 The Area Of A Triangle. Consider the right angled triangle below: 8 cm5cmWhat is the area of the triangle ?Area = ½ x 40 = 20cm2BaseHeightWhat shape is the triangle half of ?The formula for the area of a triangle is:RectangleArea = ½ x Base x HeightWhat is the area of the rectangle?A = ½ BHArea = 8 x 5 = 40 cm2
11 Does the formula apply to all triangles ? Base (B)Height (H)Can we make this triangle into a rectangle ?YesThe triangle is half the area of this rectangle:The areas marked A1 are equal.BHA1A2The areas marked A2 are equal.For all triangles:Area = ½ BH
12 Calculate the areas of the triangles below: Example 1Example 210cm6cm6.4m3.2mSolution.Solution.Area = ½ x base x heightArea = ½ x base x heightheight = 6cmbase = 10 cmheight = 3.2mbase = 6.4mArea = ½ x 10 x 6Area = ½ x 6.4 x 3.2Area = ½ x 60 = 30cm2Area = ½ x = 10.24m2
13 Example 3.Calculate the area of the shape below:Solution.16m12m10mDivide the shape into parts:A1A2Area = A1 + A2A1A210101216-12 =4Area = LB + 1/2 BHArea = 10 x ½ x 4 x 10Area =Area = 140m2
14 What Goes In The Box ? 2 Find the area of the shapes below : (1) 8cm (2)10.2 m6.3m32.13m2(3)25m18m12m258m2
15 The Area Of A Trapezium.A Trapezium is any closed shape which has two sides that are parallel and two sides that are not parallel.
16 We are now going to find a formula for the area of the trapezium: bhArea = A1 + ( A2 + A3 )Area = b x h + ½ x (a - b) x hArea = bh + ½ h(a - b)Divide the shape into parts:Area = bh + ½ ah – ½ bhA2A1A3Area = ½ ah + ½ bhWork out the dimensions of the shapes:Area = ½ h ( a + b )bA2A3A1hOften common sense is as good as the formula to work out the area of a trapezium.ha – b
17 Example 1Calculate the area of the trapezium below :16cm11cm13cmSolution ( Using the formula).Area = ½ h ( a + b )a = 16b =11h = 13Area = ½ x 13 x ( )Area = ½ x 13 x 27Area = 175.5cm2
18 16cm11cm13cmSolution ( Using composite shapes).Divide the shape into parts:Area = rectangle + triangleArea = LB + ½ BHArea = (11x 13) + ( ½ x 5 x 13 )Area =Area = 175.5cm211Decide for yourself if you prefer the formula or composite shapes.131316 – 11 = 5
19 Example 2Divide the shape into parts:8m14m10mArea = rectangle + triangleArea = LB + ½ B HA = ( 10 x 8 ) + ( ½ x 6 x 10 )A =A = 110 m 2101014 – 8 = 68
20 What Goes In The Box ? 3 Find the area of the shapes below : 13cm (1)20cm13cm10cm165cm22.7m5.4m4.9m(2)19.85m2 (to 2 d.p)
21 The Area Of A Circle. Consider the circle below divided into quarters: We are going to place the quarters as shown to make the shape belowWe can fit a rectangle around this shape:At the moment it is hard to see why this should tell us how to calculate the area of a circle.
22 Now consider the same circle split into eight parts: The eight parts are arranged into the same pattern as last time:LBThis time the shapes fit the rectangle more closely:
23 LBThis time the shapes fit the rectangle more closely:What length must the breadth B be close to ?B = rWhat length must the length L be close to ?Half of the circumference of the circle.If C = 2 r then L = r .We now have an approximate length and breadth of our rectangle.
24 What is the area of the rectangle ? A = r x rA = r 2If the circle was split into more and more smaller segments and the segments arranged in the same pattern, then the parts would become the rectangle shown above.See “Autograph Extras”, “New”, “Area Of Circle” for further info’.rConclusion.The area of a circle of radius r is given by the formula A = r 2.
25 Find the area of the circles below: Example 2Example 1.20 cm2.7mA = r 2A = r 2r = 1.35mr = 10A = 3.14 x 1.35 x 1.35A = 3.14 x 10 x 10A = 5.72m2 ( to 2 d.p)A = 314 cm2
26 Example 4Example 312cm7cmA1A27cmSplit the shape into two areas.Find half the area of a circle:Area = A1 + A2A = r 22Area = LB + ½ r 2.L = 12B = 7r = 3.5A = 3.14 x 7 x 72A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5A =A = 76.93cm2A = 103.2cm 2. (to 1 d.p)
27 What Goes In The Box ? 4 Find the area of the shapes below : (2) 6.3m (1)7cm153.86cm231.16m2 ( 2 d.p)6.7cm4.2cm(3)35.1cm 2 ( 1 d.p)