Presentation on theme: "Area Of Shapes. 8cm 2cm 5cm 3cm A1 A2 16m 12m 10m 12cm 7cm A1A2."— Presentation transcript:
Area Of Shapes. 8cm 2cm 5cm 3cm A1 A2 16m 12m 10m 12cm 7cm A1A2
What Is Area ? Area is the amount of space inside a shape: Area Area is measured in square centimetres. 1cm A square centimetre is a square measuring one centimetre in each direction. It is written as : 1cm 2
Estimating The Area. Look at the four shapes below and use your judgement to order them from smallest to largest area: A B C D
A B 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 36cm 2 each.
Area Of A Rectangle. Look again at one of the shapes whose area we estimated: C 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: Area = Length x Breadth or A = LB for short. Length Breadth
We can now calculate the area of each rectangle very quickly: (1) A= L x B A = 6 x 6 =36cm 2 (2) A= L x B A = 12 x 3 =36cm 2 (3) A= L x B A = 9 x 4 =36cm 2 (4) A= L x B A = 18 x 2 =36cm 2
Example 1 Calculate the area of the rectangle below: Solution A = LB L = 7B = 4 A = 7 x 4 A = 28cm 2 7cm 4cm (1) (2) 3m 5m This area is in square metres: 1m Solution A = LB L = 3B = 5 A = 3 x 5 A = 15m 2
Example 3. Calculate the area of the shape above: 8cm 2cm 5cm 3cm Solution. Split the shape up into two rectangles: A1 A2 Calculate the area of A1 and A2. A1 2 5 A2 3 6 Area = A1 + A2 Area = ( 2 x 5) + (6 x 3) Area = Area = 28cm 2
What Goes In The Box ? Find the area of the shapes below : (1) 8cm 6cm 4.2m 2.7m (2) (3) 17cm 8cm 12cm 5cm 48cm m 2 141cm 2
The Area Of A Triangle. Consider the right angled triangle below: What shape is the triangle half of ? Rectangle 8 cm 5cm What is the area of the rectangle? Area = 8 x 5 = 40 cm 2 What is the area of the triangle ? Area = ½ x 40 = 20cm 2 Base Height The formula for the area of a triangle is: Area = ½ x Base x Height A = ½ BH
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: B H A1 The areas marked A1 are equal. A2 The areas marked A2 are equal. For all triangles: Area = ½ BH
Calculate the areas of the triangles below: Example 1 10cm 6cm Solution. Area = ½ x base x height base = 10 cm height = 6cm Area = ½ x 10 x 6 Area = ½ x 60 = 30cm 2 Example 2 6.4m 3.2m Solution. Area = ½ x base x height base = 6.4m height = 3.2m Area = ½ x 6.4 x 3.2 Area = ½ x = 10.24m 2
Example 3. 16m 12m 10m Calculate the area of the shape below:Solution. Divide the shape into parts: A1 A2 Area = A1 + A2 A1 A =4 10 Area = LB + 1/2 BH Area = 10 x 12 + ½ x 4 x 10 Area = Area = 140m 2
What Goes In The Box ? 2 Find the area of the shapes below : (1) 8cm 10cm (2) 10.2 m 6.3m (3) 25m 18m 12m 40cm m 2 258m 2
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: a b h Divide the shape into parts: A1 A2A3 Work out the dimensions of the shapes: A1 b h A2 a – b h Area = A1 + ( A2 + A3 ) Area = b x h + ½ x (a - b) x h Area = bh + ½ h(a - b) Area = bh + ½ ah – ½ bh Area = ½ ah + ½ bh Area = ½ h ( a + b ) A3 Often common sense is as good as the formula to work out the area of a trapezium.
Example 1 Calculate the area of the trapezium below : 16cm 11cm 13cm Solution ( Using the formula). Area = ½ h ( a + b ) a = 16b =11h = 13 Area = ½ x 13 x ( ) Area = ½ x 13 x 27 Area = 175.5cm 2
16cm 11cm 13cm Solution ( Using composite shapes). Divide the shape into parts: – 11 = 5 Area = rectangle + triangle Area = LB + ½ BH Area = (11x 13) + ( ½ x 5 x 13 ) Area = Area = 175.5cm 2 Decide for yourself if you prefer the formula or composite shapes.
Example 2 8m 14m 10m Divide the shape into parts: – 8 = 6 Area = rectangle + triangle Area = LB + ½ B H A = ( 10 x 8 ) + ( ½ x 6 x 10 ) A = A = 110 m 2
What Goes In The Box ? 3 Find the area of the shapes below : (1) 20cm 13cm 10cm 2.7m 5.4m 4.9m (2) 165cm m 2 (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: This time the shapes fit the rectangle more closely: L B
L B 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.
r. r 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 cm A = r 2 r = 10 A = 3.14 x 10 x 10 A = 314 cm 2 Example 2 2.7m A = r 2 r = 1.35m A = 3.14 x 1.35 x 1.35 A = 5.72m 2 ( to 2 d.p)
A = r 2 2 Example 3 7cm Find half the area of a circle: A = 3.14 x 7 x 7 2 A = 76.93cm 2 Example 4 12cm 7cm Split the shape into two areas. A1A2 Area = A1 + A2 Area = LB + ½ r 2. L = 12B = 7r = 3.5 A = 12 x 7 + ½ x 3.14 x 3.5 x 3.5 A = A = 103.2cm 2. (to 1 d.p)
What Goes In The Box ? 4 Find the area of the shapes below : (1) 7cm (2) 6.3m 6.7cm 4.2cm (3) cm m 2 ( 2 d.p) 35.1cm 2 ( 1 d.p)