Video: Finding the Perimeter of a Composite Figure

The given figure shows two half circles and two-quarters of another circle. Find the perimeter of the shaded region, taking 3.14 as an approximate value for πœ‹.

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Video Transcript

The given figure shows two half circles and two-quarters of another circle. Find the perimeter of the shaded region, taking 3.14 as an approximate value for πœ‹.

In this question then, we have two different sizes of circle. Tracing our finger or pen all the way around the edge of the figure, we see that it is composed firstly of the arc of a semicircle. Then the arc of a different-sized quarter circle. Then the arc of a semicircle which is congruent to the first. And finally the arc of a quarter circle which is congruent to the other quarter circle.

In total then, what we have is the full circumference of the smaller circle β€” that’s the orange one β€” and half the circumference of the larger circle β€” that’s the pink one. We know that the circumference of a circle can be found using the formula πœ‹π‘‘, where 𝑑 is the diameter of the circle. So we just need to determine the diameter of each of these circles.

From the figure, we can see that the smaller circle has a diameter of 39 centimeters. So its circumference and its contribution to the perimeter of the full figure is 39πœ‹. When we think about the pink circle, however, this measurement of 39 centimeters is the radius of this circle. So the diameter is twice that. It’s 78 centimeters. The circumference of the full pink circle then would be 78πœ‹. But remember, we only have half the circumference. So the length of the semicircular arc is 78πœ‹ over two.

In fact, we find then that the two values are the same. Both the circumference of the full orange circle and the length of the semicircular arc for the pink circle are 39πœ‹. In total then, we have an exact perimeter of 78πœ‹. But looking back at the information given in the question, we’re asked to use 3.14 as an approximate value for πœ‹.

We can use a column method to work out 314 multiplied by 78, giving 24492. And we then need to divide this value by 100 to give the answer to the decimal calculation 3.14 multiplied by 78. This gives a value of 244.92. And the units for this perimeter are the same as the units given for the individual lengths in the question. They are centimeters.

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