Area Of Shapes. 8cm 2cm 5cm 3cm A1 A2 16m 12m 10m 12cm 7cm

What Is Area ? Area is the amount of space inside a shape: Area Area 1cm Area is measured in square centimetres. 1cm2 A square centimetre is a square measuring one centimetre in each direction. It is written as :

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: A B C D

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.

Area Of A Rectangle. Look again at one of the shapes whose area we estimated: C Length Breadth What was the length of the rectangle ? 9cm How many rows of 9 squares can the breadth hold ? 4 We 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 = LB for short. Area = Length x Breadth or

We can now calculate the area of each rectangle very quickly: (1) (2) A= L x B A = 12 x 3 =36cm2 (3) A= L x B A = 6 x 6 =36cm2 A= L x B A= L x B (4) A = 18 x 2 =36cm2 A = 9 x 4 =36cm2

Example 1 Calculate the area of the rectangle below: (2) 3m 5m 7cm 4cm (1) Solution This area is in square metres: 1m A = LB Solution A = LB L = 7 B = 4 L = 3 B = 5 A = 7 x 4 A = 3 x 5 A = 28cm2 A = 15m2

Example 3. Solution. 8cm 2cm 5cm 3cm Split the shape up into two rectangles: A1 Calculate the area of A1 and A2. A2 2 A1 A2 3 5 6 Calculate the area of the shape above: Area = A1 + A2 Area = ( 2 x 5) + (6 x 3) Area = 10 + 18 Area = 28cm2

What Goes In The Box ? Find the area of the shapes below : (1) 8cm 6cm (2) 48cm2 (3) 17cm 8cm 12cm 5cm 11.34m2 141cm2

The Area Of A Triangle. Consider the right angled triangle below: 8 cm 5cm What is the area of the triangle ? Area = ½ x 40 = 20cm2 Base Height What shape is the triangle half of ? The formula for the area of a triangle is: Rectangle Area = ½ x Base x Height What is the area of the rectangle? A = ½ BH Area = 8 x 5 = 40 cm2

Does the formula apply to all triangles ? Base (B) Height (H) Can we make this triangle into a rectangle ? Yes The triangle is half the area of this rectangle: The areas marked A1 are equal. B H A1 A2 The areas marked A2 are equal. For all triangles: Area = ½ BH

Calculate the areas of the triangles below: Example 1 Example 2 10cm 6cm 6.4m 3.2m Solution. Solution. Area = ½ x base x height Area = ½ x base x height height = 6cm base = 10 cm height = 3.2m base = 6.4m Area = ½ x 10 x 6 Area = ½ x 6.4 x 3.2 Area = ½ x 60 = 30cm2 Area = ½ x 20.48 = 10.24m2

Example 3. Calculate the area of the shape below: Solution. 16m 12m 10m Divide the shape into parts: A1 A2 Area = A1 + A2 A1 A2 10 10 12 16-12 =4 Area = LB + 1/2 BH Area = 10 x 12 + ½ x 4 x 10 Area = 120 + 20 Area = 140m2

What Goes In The Box ? 2 Find the area of the shapes below : (1) 8cm (2) 10.2 m 6.3m 32.13m2 (3) 25m 18m 12m 258m2

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.

We are now going to find a formula for the area of the trapezium: b h Area = A1 + ( A2 + A3 ) Area = b x h + ½ x (a - b) x h Area = bh + ½ h(a - b) Divide the shape into parts: Area = bh + ½ ah – ½ bh A2 A1 A3 Area = ½ ah + ½ bh Work out the dimensions of the shapes: Area = ½ h ( a + b ) b A2 A3 A1 h Often common sense is as good as the formula to work out the area of a trapezium. h a – b

Example 1 Calculate the area of the trapezium below : 16cm 11cm 13cm Solution ( Using the formula). Area = ½ h ( a + b ) a = 16 b =11 h = 13 Area = ½ x 13 x ( 16 + 11 ) Area = ½ x 13 x 27 Area = 175.5cm2

16cm 11cm 13cm Solution ( Using composite shapes). Divide the shape into parts: Area = rectangle + triangle Area = LB + ½ BH Area = (11x 13) + ( ½ x 5 x 13 ) Area = 143 + 32.5 Area = 175.5cm2 11 Decide for yourself if you prefer the formula or composite shapes. 13 13 16 – 11 = 5

Example 2 Divide the shape into parts: 8m 14m 10m Area = rectangle + triangle Area = LB + ½ B H A = ( 10 x 8 ) + ( ½ x 6 x 10 ) A = 80 + 30 A = 110 m 2 10 10 14 – 8 = 6 8

What Goes In The Box ? 3 Find the area of the shapes below : 13cm (1) 20cm 13cm 10cm 165cm2 2.7m 5.4m 4.9m (2) 19.85m2 (to 2 d.p)

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 below We 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.

Now consider the same circle split into eight parts: The eight parts are arranged into the same pattern as last time: L B This time the shapes fit the rectangle more closely:

L B This time the shapes fit the rectangle more closely: What length must the breadth B be close to ? B = r What 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.

What is the area of the rectangle ? A =  r x r A =  r 2 If 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’. r Conclusion. The area of a circle of radius r is given by the formula A =  r 2.

Find the area of the circles below: Example 2 Example 1. 20 cm 2.7m A =  r 2 A =  r 2 r = 1.35m r = 10 A = 3.14 x 1.35 x 1.35 A = 3.14 x 10 x 10 A = 5.72m2 ( to 2 d.p) A = 314 cm2

Example 4 Example 3 12cm 7cm A1 A2 7cm Split the shape into two areas. Find half the area of a circle: Area = A1 + A2 A =  r 2 2 Area = LB + ½  r 2. L = 12 B = 7 r = 3.5 A = 3.14 x 7 x 7 2 A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5 A = 84 + 19.23 A = 76.93cm2 A = 103.2cm 2. (to 1 d.p)

What Goes In The Box ? 4 Find the area of the shapes below : (2) 6.3m (1) 7cm 153.86cm2 31.16m2 ( 2 d.p) 6.7cm 4.2cm (3) 35.1cm 2 ( 1 d.p)