This is problem 28 of the grade 11 and 12, 2025 math kangaroo contest. The figure shows a regular octagon of side 1 centimeter. An arc of radius 1 centimeter has been drawn centrifugally at each vertex as shown. What is the perimeter of the shaded region? Our answers are A, pi centimeters, B, 2 pi over 3 centimeters, C, 8 pi over 9 centimeters, D, 4 pi over 5 centimeters, and E, 3 pi over 4 centimeters. Take your time. Try and solve this one yourself. Pause the video if you need. OK. So we know that the regular octagon is rotationally symmetric. And if we do the same thing to each vertex, then we get that the resultant thing should also be rotationally symmetric. And if you look at the picture, this is very clearly a rotationally symmetric thing. So then if we just find the length of one of these things, we know that it's the same as this one, is the same as this one, is the same as this one, and so on and so forth. Like, they're all the same. So we just need to find the length of this one and then multiply it by 8. OK. Awesome. And now let's work in maybe a bigger picture, because I've kind of ruined this one. OK. So let's do some labeling first. Let's call this arc, like this smaller arc that doesn't hit the shaded region. We're going to make that the red arc. Then the green arc is going to be the little tiny arc only on the shaded region. And then the black arc is going to be just the entire thing. OK. Awesome, awesome, awesome. So then what do we do now? Well, nothing jumps out immediately. But let's try drawing some lines in. So we can draw this line in from any vertex to the red. And then we can draw this line in. And wait, we get that these are all arcs of radius 1 centimeter. And thus, this to here is just an arc of this bigger circle. So this is 1. The side length is 1. And this is also 1. And that's an equilateral triangle. So each of these should be 60 degrees. Well, that's really, really useful. Because that also tells us, by virtue of the fact that the interior angle of an octagon. Wow, terrible act of octagon. But that's 135 degrees. And therefore, this thing here is 75 degrees. So the angle that subtends this black arc is actually just 75 degrees. So then we have 75 degrees minus the red arc's 60 degrees should equal this green arc's 15 degrees. Because the red plus the green equals the black. So we know that each little green arc is a 15 degree arc of a 1 centimeter radius circle. So then we get that if we want to calculate the perimeter, we need 8 times 15 over 360. Times 2 pi, which is just the circumference of a radius 1 circle. And then we get that this is eventually equal to 240 over 360 times pi. Which is also equal to 2 over 3 pi. And thus, our answer is b.

perimeter

shaded region

regular octagon

arcs

geometry