Grades 11-12 Video Solutions 2021-video 2021 11-12/4

Video Transcription

Problem number 4 states, A large square is divided into smaller squares as shown. A shaded circle is inscribed inside each of the smaller squares.  What proportion of the area of the larger square is shaded?  So here we have the square with some of the circle shaded in. Instead of considering the entire shape, first let's consider only one circle-square combo.  We'll say the circle inscribed in the square has a radius r. The area of the circle would be pi multiplied by the radius squared.  And the square in the circle is inscribed in has lengths twice the radius or the diameter, so mathematically 2r.  So the area of a square is the side length squared, so 2r squared is the area of the square. Now let's find the ratio of the shaded area, the circle, to the whole, the square.  So area of the circle divided by area of the square. Let's simplify the denominator. The pi's and r's cancel out, and we get pi to 4.  So for every 4 units of area, you get pi shaded area units. So that's the ratio of each individual square and circle combo. But what about the entire square?  Since all the areas have the same ratios, the overall ratio will stay the same of pi to 4. So the question asked us, what proportion of the area of the large square is shaded?  The answer is pi over 4, letter E.

Video Summary

The problem involves a large square divided into smaller squares, each containing an inscribed circle. The task is to determine the proportion of the larger square's area that is shaded by these circles. For one smaller square and its inscribed circle, the circle's radius is \( r \), resulting in an area of \( \pi r^2 \). The square's side is \( 2r \), making its area \( (2r)^2 = 4r^2 \). Thus, the ratio of the circle's area to the square's area is \( \pi/4 \). Since this ratio applies to all smaller squares and circles, the proportion of the shaded area in the larger square is also \( \pi/4 \).

Keywords

geometry

proportion

inscribed circles

area ratio

shaded area