SatsMuncher 4: Answers  The SizeWize Challenge Congratulations - your design calculations allowed Mrs Cova to completed her plans to SizeWise Secondary School ... ... but did you work out the correct sizes for each job?   DID YOU AVOID THE MOST COMMON MISUNDERSTANDING BETWEEN...?   - Perimeter is the distance - or length -  AROUND the outside of a shape?  AND... - Area is the flat space - or surface -  INSIDE the perimeter ?   Check your answers below to find out ...

Q1.  The Lower School Playground is a 5 m by 8 m.  What length of perimeter fencing does she need to order? A1.  26 m Perimeter = Total distance around the outside of the shape                 = 5 + 8 + 5 + 8   (or 2 x 5   +   2 x 8)                  = 26   Q2.  Perimeter fencing for the square Upper School playground : 36 m.  Length of each side of the playground?  A2.  9 metres Square: all 4 sides equal Perimeter = Total distance around the outside of the shape                =  Sum of 4 sides Each side = 36 / 4 = 9    (CHECK: 9 + 9 + 9 + 9 = 36  - correct)   Q3.  Lower School playground: 7 m by 5 m rectangle.  Area? A3. 35 square metres USING FORMULA FOR AREA OF ANY RECTANGLE: Area of a rectangle = Length x Width  =  7  x  5  =   35 OR VISUALISING: 5 rows with 7 squares in each row  = 5 x 7 = 35

Q4.  9C's classroom carpet: floor is a square with area of 36 square metres.  Length of each side? A4.  6 m USING FORMULA FOR AREA OF ANY RECTANGLE: Area of a rectangle = Length x Width So: Area of a square = Length x Length = Length squared ?  x  ?  =  36 6  x  6  =  36             (Or:  Length = square root of 36 = 6)   Q5.  9A's carpet is a rectangle with an area of 28 square metres. Two sides are 4 metres.  Lengths of the other two sides? A5.  7 metres each Two opposite sides are equal in rectangles. -  So, these two 4 metre sides are opposite. USING FORMULA FOR AREA OF ANY RECTANGLE: Area of a rectangle = Length x Width = L x W                       28   =   4   x   ?                       28   =   4   x   7   Q6.  Staffroom extension:  a cuboid shape, 5 m long, 3 m wide and 3 m high.  Volume?  A6.  45 cubic metres USING FORMULA FOR VOLUME OF A CUBOID (box shape): Volume = Length x Width x Height OR V   =   L  x  W x  H   =   5  x  3  x  3   =   45 OR VISUALISING: See a cuboid as rows and layers of 1 metre cubes: Bottom layer:  3 rows of 3 cubes = 9 cubic metres So: 5 layers = 5  x  9  =  45        (OR:        5  x  3  x  3  =  45)   Q7.  Sixth Form courtyard: same shape as Design A.  Which of the three designs has the same perimeter as design A?   A7.  Design D Easy mistake here was to count squares (for area) instead of lengths (for perimeter). If you count straight lengths and sloping lengths (different - slightly longer): Design A and D both have 8 straight lengths and 8 sloping lengths, so they must have the same perimeter.   Q8.  Head's rectangular car parking space. The builder arranges them into a 5 m by 6 m rectangle with an area of 30 square metres.  What other - different shaped - rectangles can you find with the same area? (There are 2 other possible rectangles.) A8.  2 new rectangles: Original rectangle: 5m by 6 m    or 6m  x  5 m 2 New rectangles:  2 m by 15 m or 15 m by 2 m                                -  (narrow for the average car)                              1 m by 30 m or 30 m by 1 m                                   -  (too narrow for any car!) Both rectangles have 30 one-metre square slabs, so both have an area of 30 square metres. Check using formula: area of a rectangle = Length x Width                                                    30   =   5       x     6                                             and  30   =   15     x     2                                                  and  30   =    1      x     30 There are no other rectangles with an integer (whole) number of slabs.  ( ... Of course, you could always hire a concrete cutter and halve some slabs:  30 =  1/2  x  60 but who drives a car only 1/2 metre wide? )

Q9.  Now it's almost the end of the spend ... but the rather shabby reception desk does not impress visitors.  Mrs Cova decides it's time for it to be replaced. The desk is triangular with an area of 10 square metres. Can you find all the possible triangles?  What sizes are they? (There are 3 different possible triangles if you stick to whole number - or integer - lengths) A9.  These three triangles with the following measurements: (b = base  and h = perpendicular -or upright - height of triangle) Formula:  'Area is half the base times the height' OR, Area   =     base   x   height   /   2 If 10   =   half of base   x   height,   then base   x   height = 20 So, it helps if if you find all of the integers (positive whole numbers) which make 20: Base Height Area = B x H / 2 20 1 20 x 1 / 2 = 10 10 2 10 x 2 / 2 = 10 5 4 5 x 4 / 2 = 10    Q10.  Finally, two very grand-looking brick pillars are built either side of the front school entrance. Each pillar is a cuboid with 10 layers of bricks.  Each layer is arranged like this:   How many bricks are in each pillar? A10.  150 bricks This layer has 3 rows of 5 bricks = 15 bricks So 10 layers has 10 x 15 = 150 bricks (OR: Length  x  Width  x  Height  =  5  x  3  x  10  =  150) The school is now finished and looks fantastic!  Mrs Cova would like to thank you for your hard work on the design calculations.