Calculate arc lengths, angles and areas of sectors of circles (1)
This is the first of two weekly sets of tasks looking at arcs and sectors. In this set we focus on the area of sectors.
MONDAY: This is a fairly routine task as it simply involves repeated applications of the procedure for finding the area of a sector (though for some of the sectors, determining the required radius and angle might be quite challenging). The task's charm lies in the fact that the two shapes turn out to have exactly the same area.
TUESDAY: Again, this is fairly routine, though as there are several steps, care needs to be taken.
A good, quick estimate of the area of the figure can be obtained by comparing it to a 3 cm by 3 cm square (below), so its area is about 9 unit squares - does it appear to be slightly more or slightly less? We can express the area (in unit squares) as this:
𝛑(1×⅜ + 2²×⅛ + 3²×⅛ + 8×⅛) = 3𝛑, which is slightly greater than 9.
WEDNESDAY: Another fairly routine task - though if the given regions do have roughly the same area why does there not appear to be a simple relation between the given angles?
If we think of the circles as having radii of 1, 2, and 3 units, we can express the areas of their respective sectors like this:
𝛑×1²×132/360, 𝛑×2²×33/360 and 𝛑×3²×15/360, which is 132×𝛑/360, 132×𝛑/360 and 135×𝛑/360.
So the green sector has a slightly larger area than the others.
THURSDAY: This might be said to complement Tuesday's task. This time one might ask, As there is a simple relation between the given angles, why does there not appear to be one for the given radii?
We can express the areas of the sectors (from left to right) like this:
𝛑×17²×30/360, 𝛑×12²×60/360 and 𝛑×10²×90/360, which is 289×𝛑/12, 288×𝛑/12 and 300×𝛑/12.
FRIDAY: This again involves the procedure for finding the area of a sector. However it is mathematically more interesting than the previous tasks, in that we have to find (and apply) a relationship rather than just make use of given values.
The areas of the circles are in the ratio 1² : 2², or 1 : 4, so the angles of the green sector and blue sector are in the ratio 4 : 1, so angle a = ⅕ of 180˚ = 36˚.
EXTRA: This is a more generic version of Friday's task [in the sense that it is 'almost'(?!) general]. You might want to use it first with your students.
