Grades 9-10 Video Solutions 2022-2022_9-10

Video Transcription

Question 11. The diagram shows three large circles of equal radius and four small circles of equal radius,  where the centers of all circles and all points of contact lie on one straight line. The radius of each small circle is one. What is the shaded area?  First, we can take a closer look at the different circles. We notice that each one larger circle contains two smaller circles,  so that means that the radius is twice as long in the large circle as it is in the small circle. Two times one is two, so the radius of a large circle will be two.  Next, we take a look at the shaded areas, and I will outline them like so, in red and in green. Now, if we wipe away the two side circles and we move over the shaded areas,  we notice that the total shaded area will be the area of one large circle minus the sum of the area of two circles.  To do this, we take the formula for the area of a circle, which is pi times radius squared. We know that the radius of a large circle is two, so pi times two squared.  The radius of a smaller circle is one, so we will subtract pi times one squared, and again, since there are two small circles.  When we simplify this, we get four pi, minus pi, minus pi. And with this, we get our solution, b, two pi.

Video Summary

The problem involves three large circles and four small circles, all centered on a straight line, with equal radii within each size group. Each large circle, containing two small circles, has a radius of two, while each small circle has a radius of one. To find the shaded area, calculate the area of one large circle and subtract the areas of two small circles from it. Using the circle area formula, \( \pi r^2 \), for a large circle (radius 2), the area is \( 4\pi \). Subtracting two small circles (\( 2\pi \)) yields a shaded area of \( 2\pi \).

Keywords

circle geometry

shaded area

circle radius

area calculation

mathematics problem