Problem
Perimeter of a figure with arcs and segments
A semicircle and an upright rectangle overlap. The shaded part is made of two pieces. The quarter circle inside the rectangle is cut away. Find the perimeter of the shaded part.
Reasoning · Grade 6-2 Circumference of a Circle
Your answer
Watch out Measuring the curves apart is fiddly, but together they make one semicircular arc.
How to solve
Strategy Identify Subproblems — A boundary made of straight bits and curved bits is really two easier problems glued together: add the straight bits with ordinary addition, and find the curved bits from the circle formula. So I first split the job that way. Before adding anything, though, I trace the outline with a finger and make a list of every piece I meet, checking each one against the rule 'shaded on one side, white on the other' - that is what stops me from including the rectangle's bottom side, which looks like an edge but has white on both sides of it. Finally I regroup the straight pieces in a smarter order and discover they rebuild the rectangle's perimeter exactly, which turns five separate lengths into one familiar formula.
1STEP 1
Name the key points from the figure
Give the key points names.
OA = OB = OP = 2 cm (all radii), OD = 4 cm, PD = 4 - 2 = 2 cm
Every segment drawn from the dot out to the arc is a radius, so it is 2 cm whichever direction it goes. Naming the points first means the rest of the work is bookkeeping instead of guesswork.
7.G.B.4Draw A Diagram2STEP 2
Trace the outline and list every piece of it
Trace the outline and list every piece.
boundary = frown{AP} + PO + OA + frown{PB} + BC + CD + DP
Perimeter is just the walk around the edge, and the honest test for 'is this an edge?' is whether the colour changes as you step across it. That single test settles the two tricky pieces at once: the bottom side is out, and the vertical radius PO is in.
3.MD.D.8Make A Systematic List3STEP 3
Add the curved parts: they make one full semicircular arc
The two curves add to 6.2 cm.
(circumference) = 2 × 2 × 3.1 = 12.4 cm, frown{AP} + frown{PB} = 12.4 ÷ 2 = 6.2 cm
There is no need to work out each quarter arc separately and add them; the two quarters are back to back on the same circle, so they add up to a half circle, and half of the circumference is a single easy calculation.
7.G.B.4Identify SubproblemsThe curved pieces of the outline join up into exactly one semicircular arc.
Why?
The pieces meet end to end with no gap and no overlap, so together they are one continuous arc rather than several.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
All the curved pieces come from circles of the same radius, so they can be added as arcs of one circle.
🧱A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.
4STEP 4
Add the straight parts, then regroup them into the rectangle
The straight pieces regroup into the rectangle's 12 cm.
2 + 2 + 2 + 2 + 4 = 12 cm = (2 + 4) × 2
Rearranging the same five lengths into a shape you already know turns a five-term addition into one formula and gives a free check on the arithmetic. It works because the missing bottom side and the extra radius on the left are both 2 cm - the very fact the figure is built on.
4.MD.A.3Organize Information In More Ways5STEP 5
Put the straight total and the curved total together
Together that is 18.2 cm.
(perimeter) = (2 + 4) × 2 + 2 × 2 × 3.1 ÷ 2 = 12 + 6.2 = 18.2 cm
Only the curved part brings in a decimal, so the final sum is a whole number plus one decimal place - line up the decimal points and the addition is a single step.
5.NBT.B.7Identify SubproblemsAnswer
18.2 cm
12 + 6.2 = 18.2
The units are right: every piece added was a length in centimetres, and no area ever entered the calculation. The size is sensible too. A rough upper estimate: the whole figure fits inside a box 4 cm wide and 4 cm tall, whose perimeter is 16 cm, and the shaded outline is a little longer than that box because the arc bulges, so a number just above 16 is expected - 18.2 fits. A rough lower estimate: the rectangle alone has perimeter 12 cm, and the outline clearly adds a big curved detour, so it must be well above 12. Also, the semicircular arc had to be longer than the straight diameter it spans, and 6.2 cm is indeed longer than 4 cm, which is a good sign that the circle part was not miscalculated. The two easy traps were both avoided: the rectangle's bottom side was correctly left out, and the vertical radius from the centre up to the top of the arc was correctly put in.
Takeaway
Walk the outline with your finger: the straight bits rebuild the rectangle (12 cm) and the curved bits make half a circle (6.2 cm), so the trip is 18.2 cm.
- Name the key points from the figure
- Trace the outline and list every piece of it
- Add the curved parts: they make one full semicircular arc
- Add the straight parts, then regroup them into the rectangle
- Put the straight total and the curved total together