Reasoning · Grade 6-2 Area of the Shaded Region
Problem
A band is a difference of two sectors.
This is the same trick as finding the area of a picture frame: measure the whole thing including the hole, then take the hole away. Nothing new is needed beyond the sector formula you already have.
6.G.A.1Identify SubproblemsEach curved band is the big sector with the small sector taken out of it.
The band and the small sector fill the big sector with no gap and no overlap, so their areas add back to it.
Knowing the whole and the part taken out, subtracting returns the band, because taking away undoes putting together.
In (1) the radii are 10 m and 20 m.
The second label is a width, not a radius. Adding it to the first label is the only step in the whole problem where the picture, rather than a formula, decides the numbers.
7.G.B.4Draw A Diagram45 degrees is an eighth of a circle.
A whole circle is easier to picture than a wedge. Once you know the wedge is one eighth of the circle, the sector formula is just the circle formula with a fraction stuck on the end.
4.NF.B.4Solve An Easier Related ProblemSubtracting, (1) is 117.75 m².
Both sector areas come out of exactly the same formula, so the only real work is doubling the radius and squaring it. Subtracting decimals to hundredths is Grade 5 arithmetic.
7.G.B.6Identify SubproblemsIn (2) the radii are 6 m and 12 m.
Reading the two 6 m marks as inner radius and width, rather than as two radii, is what keeps the outer radius at 12 m. The picture shows the outer arc twice as far from the centre as the inner arc, which agrees.
7.G.B.4Draw A DiagramSubtracting, (2) is 113.04 m².
Same recipe as figure (1), only the numbers change. Doing the identical steps twice is what makes the pattern in the next step visible.
7.G.B.6Identify SubproblemsMidline times width gives the same.
Doing two problems the long way and then noticing they both equal (midline) x (width) is how a shortcut earns your trust. The inner arc is too short and the outer arc is too long, so the arc exactly in the middle is the fair one to use.
6.NS.B.3Look For A PatternA curved path is just a big wedge with a smaller wedge taken out of the middle - and its area is always the middle arc times the width.