Problem
Perimeter of a sector and of a sector ring
A sector with a 40 degree central angle is drawn. Only a band of constant width near its outer end is shaded. It is like a slice cut from a circular ring. Find the perimeter of the shaded part.
Reasoning · Grade 6-2 Circumference of a Circle
Your answer
Watch out Adding just the two arcs misses the two straight edges.
How to solve
Strategy Draw a Diagram — Perimeter questions on curved figures are won or lost at the tracing stage, so I first put my finger on the shaded region and walk all the way around it, naming each piece I cross. That splits one hard question into four easy ones: two arcs and two straight segments. For each arc I use an easier related problem first — how long is the WHOLE circle of that radius? — and then take the 40/360 share of it. Adding the four pieces gives the answer.
1STEP 1
Trace the boundary and name its four pieces
The boundary is two arcs and two straight edges.
perimeter = (inner arc) + (outer arc) + 9 + 9
Perimeter always means the same thing, curved or not: the distance you walk going once around the outside. Tracing with a finger keeps you from forgetting a piece or counting a line that is only a helper line.
3.MD.D.8Draw A Diagram2STEP 2
Read the two radii off the figure
The radii are 36 cm and 45 cm.
r_inner = 36 cm, r_outer = 36 + 9 = 45 cm
Every arc length depends on the radius of the circle the arc came from, so before any calculating you must know exactly which two circles the two arcs belong to.
7.G.B.4Draw A Diagram3STEP 3
Ask an easier question first: what fraction of a circle is 40 degrees?
40 degrees is a ninth of a circle.
40°/360° = 40/360 = 1/9
A whole circle is much easier to think about than a piece of one. Once you know the piece is one ninth of the whole, arc length is just a fraction of a number you already know how to find.
4.NF.B.4Solve An Easier Related ProblemA 40 degree sector is one ninth of its whole circle, so its arc is one ninth of the circumference.
Why?
One complete turn around the centre is 360 degrees, so 40 degrees is a definite share of that whole turn.
🧱Angles around a point make 360 degreesAngles that fit side by side all the way around one point fill exactly one full turn, 360 degrees.
Why?
Nine such sectors would fill the circle with no gap and no overlap, so each one holds one of nine equal shares of it.
🧱A fraction is equal shares of one wholeThe bottom number names how many equal pieces one whole was cut into and the top counts them, which is why same-sized pieces add just by counting.
4STEP 4
Find the inner arc
The inner arc is 24 cm.
36 × 2 × 3 × 40°/360° = 216 × 1/9 = 24 cm
Circumference is (radius) x 2 x pi. With pi taken as 3, the arithmetic stays whole-number friendly: 216 divided by 9 is 24, no decimals needed.
7.G.B.4Identify Subproblems5STEP 5
Find the outer arc
The outer arc is 30 cm.
45 × 2 × 3 × 40°/360° = 270 × 1/9 = 30 cm
Same angle, bigger radius, longer arc. The two arcs differ by 30 - 24 = 6 cm, and 6 is one ninth of the extra circumference that the 9 cm of extra radius buys you.
7.G.B.4Identify Subproblems6STEP 6
Add the two straight edges and total everything
Adding the straight edges gives 72 cm.
24 + 30 + 18 = 72 cm
Once each piece has been measured, perimeter is nothing but addition. Grouping the curved parts together and the straight parts together makes the sum easy to check.
3.MD.D.8Identify Subproblems7STEP 7
Check the units
Every piece is a length, so it checks.
Checking units catches the most common slip on ring problems, which is accidentally computing an area when a perimeter was asked for.
7.G.B.4Analyze The UnitsAnswer
72 cm
24 + 30 + 18 = 72
The answer is a length in centimetres, as a perimeter must be. It is also the right size: the shaded band roughly fits inside a rectangle about 30 cm long (the arcs) by 9 cm wide, and such a rectangle has perimeter about 2 x 30 + 2 x 9 = 78 cm, close to 72 cm. Each arc is sensible on its own too: the inner arc, 24 cm, is a ninth of the 216 cm circumference of a 36 cm circle, and the outer arc, 30 cm, is a ninth of the 270 cm circumference of a 45 cm circle, so the outer arc is longer than the inner one, exactly as the picture shows. Finally, both arcs are far shorter than their radii times four, which is what you would expect from a narrow 40 degree slice.
Takeaway
Walk your finger around the shaded part first: two arcs and two 9 cm edges are all there is, and each arc is just one ninth of its own circle.
- Trace the boundary and name its four pieces
- Read the two radii off the figure
- Ask an easier question first: what fraction of a circle is 40 degrees?
- Find the inner arc
- Find the outer arc
- Add the two straight edges and total everything
- Check the units