SESSION 6: PERIMETER, AREA AND VOLUME KEY CONCEPTS: Perimeter and Area - Rectangle -Triangle - Circle Surface Area and Volume - Rectangular prism - Triangular prism - Cylinder X-PLANATION Units of Measurement Perimeter is always 1 dimension and the units are: mm, cm or m Area is always 2 dimensional and the units are mm 2, cm 2 or m 2 Volume is always 3 dimensional and the units are mm 3, cm 3 or m 3 Perimeter and Area: a) Rectangle Perimeter of a rectangle = 2(L + B) Area of rectangle = length x breadth = L x B Example: If L = 8m and b = 4m, calculate the perimeter and area of the rectangle. Perimeter = 2(L + B) Area = LB P = 2 (8+4) = 2 (12) = 24m Triangle A = 8 x 4 = 32m 2 Perimeter of a triangle = all three sides added (P = b + s 2 + s 3 ) Area of a triangle = ½ x base x perpendicular height = ½ x b x h Page 32
Example: If b = 6m; h = 4m; s2 = 5m and s3 = 5m, calculate the perimeter and area of the triangle. P = s 1 + s 2 + s 3 A = ½ x b x h = 6m + 5m+ 5m = 16m c) Circle Radius of a circle is the diameter 2 Pi (π) = 3,14 Perimeter of a circle (circumference) = 2 x π x radius = 2πr = πd Area of a circle = π x radius squared = πr 2 Example: If r = 5m, calculate the circumference and area of the circle = ½ x 6 x 4 = 12m 2 P = 2πr A = πr 2 = 2 x 3,14 x 5m = 31,4m = 3,14 x (5m) 2 = 78,5m 2 Page 33
Surface Area and Volume a) Rectangular prism Surface area = 2(l x b + l x h + b x h) Volume = l x b x h Example: If l = 4m; b = 3m and h = 1,5m SA = 2(l x b + l x h + b x h) V = l x b x h SA = 2(4m x 3m + 4m x 1,5m + 3m x 1,5m) V = 4m x 3m x 1,5m SA = 45m 2 V = 18 m 3 b) Triangular prism Surface area = 2( x b x h) + s1 x H + s2 x H + s3 x H Volume = x b x h x height of prism (H) Example: If b = 6m; h = 4m; s2 = 5m; s3 = 5m and H = 3m SA = 2( x b x h) + s1 x H + s2 x H + s3 x H V = x b x h x H SA = (6m x 4m + 6m x 3m + 5m x 3m + 5m + 3m V = x 6m x 4m x 3m SA = 72m 2 V = 36m 3 Page 34
c) Cylinder Surface area = 2 x π x r 2 +2 x π x r x H Volume = π x r 2 x H Example: If r = 3m and H = 4m SA = 2 x π x r 2 +2 x π x r x H V = π x r 2 x H SA = 2 x π x 3 2 m +2m x π x 3m x 4m V = π x 3 2 m x 4m SA = 131,95m 2 V = 113,1m 3 X-AMPLE QUESTIONS: Question 1: The diagram is a plan for a mosaic designed for an art competition. The diameter of the circle is 5,66cm. The sides of the square and the hypotenuse of the triangles are all 4 cm long. The other sides of the right angled isosceles triangles are all 2,83 cm long. Page 35
a.) c.) d.) e.) The purple square is made from a nylon chord. How long is the chord? What is the circumference of the circle? Find the total area of the triangles Estimate the perimeter of the mosaic The diagonal of the mosaic is 9,66 cm long. Calculate the area the mosaic covers. Question 2 Study the floor plan of Phil s kitchen below (not drawn to scale) and answer the questions that follow: a) Calculate the area of the floor. b) Phil decides to replace the linoleum flooring in his kitchen with tiles. The tiles that he has chosen are square measuring 40cm by 40cm. i) Calculate the area of each tile in m 2. ii) Calculate the minimum number of tiles Phil will need to tile his kitchen based on your area calculations. iii) When Phil speaks to the tiler, he uses the length and breadth of the title and the kitchen to estimate the number of tiles he needs. Explain why this is a more practical calculation. c) In addition to putting down new tiles, Phil will need to put new skirting boards all the way around the edge of the kitchen floor, except for the three entrances. Each entrance is 1,55 m wide. i) Calculate the perimeter of Phil s kitchen. ii) Skirting board comes in lengths that are 3m long. How many lengths will Phil need? Page 36
Question 3 Patrick makes a model of a house out of cardboard boxes. He selects a rectangular prism for the base and triangular prism for the roof. He glues the triangular prism on top of the rectangular prism. He secures his house onto a flat board. He decides the paint the walls of the model white and the sloping roof red. a.) Calculate the area of the model that he paints red Calculate the area of the model that he paints white Question 4 Mr Patel owns a shop. He ordered two boxes of jam from his wholesaler. Each box contains 48 tins of jam. One of the boxes broke so he needs to repack the 48 tins. He does not have a box the same size as the original box but has two smaller boxes that he thinks he can use to pack in all the tins. Information Dimensions of jam tin: Diameter = 7cm Height = 11cm Packing in original box 6 rows of 4 tins by two layers Volume of box = 33 750 cm 3 Dimensions of box 1 Base = 25 cm x 25 cm Height = 30 cm Dimensions of box 2 Base = 36 cm x 20 cm Height = 25 cm a.) c.) d.) e.) Estimate the dimensions of the original box Calculate the volume of one tin of jam Calculate the volume of the two boxes Mr Patel wants to use to repack the tins into Use the volume of the two boxes and the volume of a jam tin to decide how many tins can theoretically fit into each box Do you think Mr Patel will be able to pack all the tins into these two boxes? Suggest how the tins should be arranged Page 37