Area of Plane Shapes 1

Transcription

1 Area of Plane Shapes 1 Learning Goals Students will be able to: o Understand the broad definition of area in context to D figures. o Calculate the area of squares and rectangles using integer side lengths. Area Introduction The area of a shape is the amount of flat surface enclosed by the shape. Area is measured in square units such as; square millimetres (mm ), square centimetres (cm ), square metres (m ), square kilometres (km ) etc... If a shape is drawn on 1-centimetre grid paper, its area can be found by counting the number of squares that the shape covers. Example 1 ~ Count the number of squares to find the area of the shape drawn on this centimetre grid. a) b) cm cm Area of Squares & Rectangles A square is a plane figure with four equal straight sides and four right angles. o The area of a square can be calculated by multiplying two adjacent sides together. o Area (A) = side side s OR s s s OR s Example ~ Find the area of the square figures below: a) b) 4 mm 4 mm 8 m mm m

2 A rectangle is a plane figure with four straight sides and four right angles, especially one with unequal adjacent sides, in contrast to a square. It is important to note that every square is classified as a rectangle but not every rectangle will be considered a square. o Area (A) = base height height OR b h base NOTE: The area formula for a rectangle can use different names for sides (e.g. length and breadth) Example 3 ~ Find the area of the rectangular figures below: a) b) 6 mm 11 m 9 mm 4 m mm m

3 Grid Area of Shapes FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE Find the area for the shapes on 1 cm grid 1 cm² Use grid overlay to find the area of these, or use a rule to make 1 cm spaced lines Use 1 cm grid to find the area of these, then measure the side lengths. Answer the questions without drawing the shapes. Calculate the area of.. 10 cm² 14 cm cm Side = Side = cm² cm cm 17 A rectangle with side lengths 8 cm and 6 cm. Area = cm cm Area = cm² Multiply sides 18 A rectangle with side lengths 7 cm and 9 cm cm cm Side = Side = 16 cm cm² cm cm Multiply sides Area = cm cm Area = cm² 19 A square with side lengths of 9 m. Area = m m Area = m² 0 A square with side lengths of 7 cm. cm Area = cm cm Area = cm² 13 Side = Side = cm² cm cm Multiply sides 1 A rectangle with side lengths 19 m and 3 m. Area = m m Area = m²

4 1 Find the areas of the squares below l ² 5 m = 0 m² Area of Squares FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE Find the areas of the squares below 6 1 cm 11 Now instead of a diagram read the problems A quilt is to be made of 19 cm side length square patches. Calculate the area of one of these patches cm 16 mm 1 A garden shed has a square roof with 13 m side lengths. Calculate the area of the roof in square metres m 8 cm 13 A courtyard has a square grassed section with 7 m long sides. Find the area for this section mm 17 cm 5 3 cm m 14 If turf costs $4/m² what would be the cost of relaying the grass surface above? Cost = area unit cost

5 Find the areas of the rectangles below Area of Rectangles FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE 1 3 mm 6 mm Find the areas of the rectangles below 6 7 m 13 m 11 Now find the area in the following problems A rectangular coffee table has side lengths 6 m and 3 m. Calculate the area of the table bh = 0 1 cm mm² 7 9 cm 45 cm 16 cm 1 Find the area of an air hockey table which has a rectangular playing area of side lengths 10 cm and 6 cm. 3 3 m 8 6 cm 5 m 14 cm 13 A foyer of a building is to be tiled. If the area is rectangular with sides 5 m and 48 m, find the area cm 4 cm 34 mm 4 mm 5 9 mm m 14 If tilers can lay 80 m² per day, calculate the time taken to complete the job 74 mm 43 m

6 Learning Goals Students will be able to: Area of Plane Shapes o Find the area of a triangle using the formula 1 bh o Understand that the area of a triangle is derived from halving the area of a rectangle Area of a Triangle 1) Deriving the area of a triangle a. If you draw a diagonal on a rectangle, it is split up into two triangles with equal area as shown in diagram below b. The area of a rectangle is bh. Since a triangle is half a rectangle, its area must be 1 bh c. The following graphic demonstrates why this formula works for different triangles: ) Use the following examples or come up with some on your own to practise calculating the area of a triangle. Example 1 ~ 1 bh 1 units

7 Example ~ 1 bh 1 units Example 3 ~ 1 bh 1 units Example 4 ~ A triangle has a base of 60 cm and a height of 5 cm. What is its area? 1 bh 1 cm Extension Example ~ The area of a triangle is 4 m and its base is 6 m. Find its height.

8 Select the measurements to use, circle them, then find the triangle's area mm 8 mm Area of Triangles FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK 1 - LICENSED FOR NON-COMMERCIAL USE 6 4 mm Find the area now using decimals 8 mm 11 Now find the area in the following problems A triangular sun shade has perpendicular sides of 5 m and 6 m. Calculate the area 6 mm ½bh = ½ mm² 13 m 7 4 m 1 m 5 m m 1 A triangular section of glass has a base of 4 m and a perpendicular height of 7 m. Calculate the area cm 6 cm cm 14 cm 8 9 cm 4 cm 13 A triangular sail has vertical height of 5 m and a horizontal base length of 3 m. Calculate its area m 8 m 10 m 8 m 9 1 mm 3 mm 5 15 cm 8 cm 10 9 m 14 A triangular garden bed has perpendicular sides of 86 cm and 94 cm. Calculate the bed's area. 17 cm 9 m

9 Area of Plane Shapes 3 Learning Goals Students will be able to: o Practically demonstrate the similarities between different plane shapes: Specifically the area of a rectangle and parallelogram. o Calculate the area of plane shapes: Specifically the area of parallelograms, kites and rhombuses with integer values Compounding Prior Learning Quick revision of calculating the area of a rectangle. Example 1 ~ base height 8 cm cm 80 cm Area of Parallelograms Video (1 min) ~ Miss Norledge - Area of a parallelogram from a rectangle (cut-out demo): You may play this a second time if appropriate for your class. Example ~ base height 6 cm 7 cm cm 10 cm Highlight the two perpendicular lengths placing emphasis on the right angle symbol. 1 cm base height cm

10 Area of Rhombuses & Kites Both rhombuses and kites don t have outside edges that are perpendicular. However, the diagonal lengths in both of these shapes are perpendicular. Multiplying these together produces a larger rectangle twice the size of the original shape, hence we halve the result. Area = 1 xy (where x and y are the lengths of the diagonals.) y x Example 3 ~ 1 diagonal diagonal 1 cm

11 Area of Parallelograms and Rhombuses FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK - LICENSED FOR NON-COMMERCIAL USE Example Find the area of these parallelograms. Round to 1 d.p. if needed. 1 bh 10 cm 8 cm 8 cm 17 cm bh = cm 7 m 6 m mm A rhombus is like a square that has been pushed over. Find the area, 1 d.p. when needed. Example 47 mm 9 10 ½ xy 11 cm 18 cm ½ xy Calc: = = ½ mm 14 mm 0 mm ½ cm cm mm 1 4 m 8 mm 11mm 4m 5 m 1 m m 7 m Find the area in cm. 5 1 cm 15 cm 6 0 mm 9mm All units are in cm m 6 cm 8 3 m m

12 Area of Kites FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK - LICENSED FOR NON-COMMERCIAL USE 1 Find the area of these kites. Round to 1 d.p. if needed. ½ xy Calc: = 10 cm = ½ cm 5 cm Example 3 m 16 cm 1 cm 1 m ½ xy ½ cm 3 4 7cm 4 mm 8 cm 6 mm From Q.5 all units are in cm

13 Area of Plane Shapes 4 Learning Goals Students will be able to: o Identify a trapezium as a quadrilateral that has one pair of parallel sides o Find the area of isosceles trapezia and other, irregular trapezia Area of a Trapezium 1) Defining a trapezium a. A trapezium is a four sided shape (quadrilateral) that has one pair of parallel sides. All the shapes below are trapezia: b. The area of a trapezium is given by the following: 1 h(a + b) where a and b are the parallel sides. c. You can also find the area of a trapezium by making it two triangles as seen below: ) Use the following examples or come up with some on your own to practise calculating the area of a trapezium. Example 1 ~ 1 h(a + b) 1 ( + ) cm

14 Example ~ 1 h(a + b) 1 ( + ) cm Check if your answers are correct by splitting up the trapezium into triangles and adding their areas together. Example 3 ~ A trapezium has a perpendicular height of 4 cm and parallel side lengths of 8 cm and 1 cm. What is its area? 1 h(a + b) Extension Example ~ 1 ( + ) cm A trapezium has an area of 48 cm and parallel sides of length 7 cm and 5 cm. What is its perpendicular height?

15 Area of Trapeziums FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK - LICENSED FOR NON-COMMERCIAL USE Find the area of the trapezia below. Dont forget square units. Eg. mm, cm, m. Round to 1 d.p. Example 5 cm ½ h (a+b) 9 A trapezium shaped courtyard has a perpendicular height of 3 m and the pair of parallel sides lengths are m and 4 m. What is the area of the courtyard? 1 3 cm 10 cm 4 cm ½ h (a+b) = ½ ( 4 cm 5 cm cm = ½ 4 ( ) 30 + ) cm 1 cm cm 3 cm 10 A sports field is in the shape of a trapezium. A coach needs to know the playing surface area for some training drills. The perpendicular height of the field is 30 m and the pair of parallel sides are 0 m and 10 m. a) Calculate the area of this field. 3 7 mm 4 mm 8 mm 4 8 m 11 m m b) If turf costs $5 per square metre, calculate the cost of resurfacing this playing field. 5 1 cm 8 cm 10 cm m 4 m 5 m 15 m 11 The area of a trapezium is 14 m and the two non-parallel sides are equal in length and 6 m. If the perpendicular height is 8 m and the perimeter is 43 m, what is the sum of the two parallel side lengths? 8 m 10 m 8 40 cm 70 cm Answer in cm 1 m

16 Area of Plane Shapes 5 Learning Goals Students will be able to: o Find missing side lengths on composite shapes o Find the area of composite shapes comprising of two or more shapes Area of Composite Shapes 1) Defining composite shapes a. Composite shapes are shapes that are comprised of two or more shapes. To help understand what this means, think of a composite word like sandbox. There are two words, sand and box, that make the one word. ) Use the following examples or come up with some on your own to practise calculating the area of composite shapes. Example 1 ~ A 1 = = cm A = = cm A Total = + = cm Example ~ A 1 = = cm A = 1 = cm A Total = + = cm Example 3 ~ A 1 = = m A = = m A Shaded = = m

17 Example 4 ~ A 1 = 1 = m A = = m A Shaded = = m

18 Area of Composite Shapes - Non Circular FREEFALL MATHEMATICS - FREEFALL MATHEMATICS ALTITUDE BOOK - LICENSED FOR NON-COMMERCIAL USE Composite areas are found by dividing the shape into two or more plane shapes, finding their areas then adding or subtracting them. 1 6 cm 3 cm A 1 5 cm A 1 = bh = A 1 = cm 5 7 m 8 m 13 m 1 A 3 cm A = bh A 1 +A = + = 6 Find the shaded area. cm A = cm 14 cm 7 m 1 m A 1 A 5 m A 1 = = A 1 = m A = 8 cm cm A 1-3A = A 1 - A = - m = A = m 7 44 m 3 3 mm 5 mm 1 m mm 3mm 4 Find the shaded area. 1 mm 8 mm 10 Find the shaded area. mm mm mm 84 mm mm 31 mm 15 mm

19 Paper Land Ships Objectives: 1. Create a paper ship using plane shapes that will glide atop several desks.. Calculate the area of your land ships sail and base. 3. Your land ship must be able to be propelled by a single fan. Equipment: Coloured paper (you can pick as many as you want) 15 cm of tape Scissors Glue sticks Ruler Activity: You will be required to make a paper land ship. You must abide by the following rules. You will be working in pairs You can create your land ship by o Folding one plane shape (the shape can be composite) o Attaching two plane shapes together. One for the sail, one for the base (the shape can be composite) You will need to fill out your planning proforma before building your land ship After the 0 minutes are complete, we will see which ship can be blown across tables the fastest.

20 Planning Proforma Draft 1 Front Back Draw a neat sketch of your land ship from different viewpoints in the space provided Side Bottom What shape are we using for the base of our land ship? What is the area of this shape? What shape are we using for the sail of our land ship? What is the area of this shape? Construct the first draft of your land ship, test it, then come back and write some things you might need to fix or change in the space provided. Use this for your second and final draft

21 Planning Proforma Draft Front Back Draw a neat sketch of your land ship from different viewpoints in the space provided Side Bottom What shape are we using for the base of our land ship? What is the area of this shape? What shape are we using for the sail of our land ship? What is the area of this shape?

22 Reflection Questions 1. Evaluate the effectiveness of your land ship.. How do you think the area of the base affected how fast the land ship travelled? 3. What shape would you use for the base if you could do this activity again? Why would you choose that shape? 4. How do you think the area of the sail affected how fast the land ship travelled? 5. What shape would you use for the sail if you could do this activity again? Why would you choose that shape?