We received concise statements of general rules based on clearly presented evidence: Show Matthew, Holly and Isobel from Staunton & Corse C of E School discovered the relationship between the enlargement scale factor and the effect on the area: We started by writing 5 different dimensions with an area of $20$cm$^2$: Length x Width: 0.25 x 80, 0.5 x 40, 1 x 20, 2 x 10, 4 x 5 Then we enlarged them by a scale factor of 2: 0.5 x 160, 1 x 80, 2 x 40, 4 x 20, 8 x 10 We then took the areas of both rectangles ($20$cm$^2$ and $80$cm$^2$) and figured out a way that 20 could be turned into 80. Well, obviously you multiply it by 4. But if you enlarge them by a scale factor of 3: 0.25 x 80, 0.5 x 40, 1 x 20, 2 x 10, 4 x 5 become: 0.75 x 240, 1.5 x 120, 3 x 60, 6 x 30, 12 x 15 This time the areas are $20$cm$^2$ and $180$cm$^2$ which means that the areas have been multiplied by 9. So the question we asked ourselves was: Akintunde from Wilson's School generalised for two and three dimensions: A rectangle with area $20$cm$^2$ could have the following dimensions: Length by Width: When enlarged by a scale factor of 2 the dimensions become: 8 by 10, 4 by 20, 40 by 2, 80 by 1, 100 by 0.8 All these rectangles have an area of $80$cm$^2$ The area is four times larger. A rectangle with an area of $10$cm$^2$ could have the following dimensions: 5 by 2, 1 by 10, 20 by 0.5 When enlarged by a scale factor of 2 the dimensions become: 10 by 4, 2 by 20, 40 by 1 All these rectangles have an area of $40$cm$^2$ Again, the area is four times larger than the original. I think that maybe whatever scale factor you enlarge the rectangle by, the area is enlarged by the square of the scale factor: SF 2, AREA increases by 4 SF 3, AREA increases by 9 Starting with the rectangles with an area of $10$cm$^2$: 5 by 2, 1 by 10, 20 by 0.5 When they are enlarged by a scale factor of 3, the dimensions become: 15 by 6, 3 by 30, 60 by 1.5 All these rectangles have an area of $90$cm$^2$ The areas are nine times larger. And when the rectangles with an area of $10$cm$^2$ are enlarged by a scale factor of 4 the dimensions become: 20 by 8, 4 by 40, 80 by 2 All these rectangles have an area of $160$cm$^2$ The areas are sixteen times larger. And when the rectangles with an area of $10$cm$^2$ are enlarged by a scale factor of 0.5 the dimensions become: 2.5 by 1, 0.5 by 5, 10 by 0.25 All these rectangles have an area of $2.5$cm$^2$ The areas are four times smaller (you multiply the original area by 0.25). I think that for any scale factor, you square it and then multiply that new number by the original area. I think this happens because the two dimensions of a rectangle are both multiplied by three when the rectangle is increased by a Scale Factor of 3 so overall the area of the rectangle is nine-times bigger (3 x 3). If you increase a rectangle by a scale factor k, the area of the new rectangle will be k$^2$ times the old area of the rectangle. Taking a triangle with a base of 2, a height of 3 and an area of 3: When enlarged by a scale factor of 2 it becomes a triangle with a base of 4, a height of 6 and an area of 12. The area is 4 TIMES LARGER. When enlarged by a scale factor of 3 it becomes a triangle with a base of 6, a height of 9 and an area of 27. The area is 9 TIMES LARGER. When enlarged by a scale factor of 4 it becomes a triangle with a base of 8, a height of 12 and an area of 48. The area is 16 TIMES LARGER. I think this rule applies to all plane shapes because when different rectangles and triangles are increased by different scale factors, the increase in area is the same. Taking a cuboid with a width of 2, a length of 2, a height of 3, a surface area of 32 and a volume of 12: When enlarged by a scale factor of 2 it becomes a cuboid with a width of 4, a length of 4, a height of 6, a surface area of 128 and a volume of 96. The surface area is 4 TIMES LARGER and the volume is 8 TIMES LARGER. The surface area increased by 4 (2$^2$) and the volume increased by 8 (2$^3$). The volume takes in 3 dimensions and the surface area takes in 2 dimensions. I have
concluded that with rectangles and cuboids, whatever the scale factor is: Based on the fact that the rule about scale factors seems to work for all plane shapes, I predict that 3D shapes will follow the same rule as for cuboids. Nice work - well done to you all. Why the area of a rectangle increased by a factor of 4 when the side length is doubled?The explanation behind the area of a rectangle increases by a factor of 4 when the side length is doubled is due to the fact that; The length and width are double which translates to a product of 4 which means the area will increase by a factor of 4.
What will happen to the area of a rectangle if its side is doubled?Therefore, area remains same.
What happens to area when length is doubled?Hence, area of rectangle become 2 t i m e s when length is doubled and breadth remains same. (i) Length and breadth are trebled. (ii) Length is doubled and breadth is same.
What is the effect on the area of a rectangle if its length is doubled and its width is tripled?Hence, the area of rectangle becomes 6 times more than its original area.
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