in this video we're going to talk about how to calculate the area of the Shader region so let's say if we have a square inside of a rectangle and let's say the side length of the square is 5 inches and the rectangle let's say it's 8 by 10. calculate the area of the Shader region to calculate the area of the shade of region you need to find the difference between the area of the large object and the area of the small object so in this example the area of the shade of each and is going to be the area of the rectangle minus the area of the square the area of the rectangle is the left times the width the area of the square is sine squared so in this case the left of the rectangle is 10 the width is 8 and S is 5 for the square so it's ten times eight minus 5 squared so the area of the rectangle is 80. the area of the square is 25. and 80 minus 25 is 55. so the area of the Shader region is 55 square units or inches squared now let's try another example so let's say if we have a square and there's a circle inside of the square and let's say the radius of the circle is eight units and let's say the square is 20 units each side with this information calculate the area of the Shaded region feel free to pause the video if you want to so we have the side length of the square and a radius so the area of the Shader region is the area of the larger object which is the square minus the area of the smaller object which is the circle now we know the area of the square is s squared the area of the circle is pi r squared so this is going to be 20 squared minus pi times H squared now 20 squared or 20 times 20 that's 400. 8 times 8 is 64. so the exact answer is 400 minus 64 pi now let's get this answer in terms of a decimal and so this comes out to be 198.9 square units so that's the answer let's try another simple problem before we start working on the harder problems so let's say there's a circle within another Circle and the inner radius let's say it's four units long and let's say the outer radius is seven units long so with this information calculate the area of the Shader region so the area of the Shaded region is the difference between the area of the large Circle minus the area of the small circle so for the large Circle it's going to be pi r squared by using the outer radius and for the small circle it's pi r squared but using the inner radius so the outer radius is 7. the inner radius is 4. 7 squared is 49 4 squared is 16. so it's 49 Pi minus 16 pi and that's going to be 33 pi so that's the area of the Shaded region that's the exact answer now let's try some harder problems so let's say this is the center of the circle and let's call this a b and c so B is the center and it's also a right angle and let's say the radius of the circle is a what is the area of the shade region the area of the Shader region is going to be the difference between the area of the large object which is the circle minus the area of the small object which is the triangle the area of a circle is pi r squared and the area of a triangle is one half base times sine so we have the radius notice that a b is the radius and also BC is the radius as well so a b and b c is eight the distance between the center and any point on a circle is the radius of the circle now notice that the radius is also the base of the triangle and it's also the height of the triangle as well so 8 squared is 64. and half of 64 is 32. so this is the area is 64 pi minus 32. and so as a decimal that's 169.1 square units so that's the answer here's another example so let's say if we have a rectangle and inside a rectangle we have a rhombus now let's say this side of the rectangle is eight and this part of the rhombus is five what is the area of the Shader region go ahead and try this problem so the area of the Shader region is the difference between the area of the rectangle and the area of the rhombus so let's try the diagnose of the rectangle the area of a rectangle is length times width the area of a rhombus is one half of D1 times D2 so D1 is basically the same as the width of the rectangle and D2 is the same as the length of the rectangle in this example so I'm going to replace D1 with W and D2 with l so LW minus one half LW will just be one half LW so we have the width already W is eight what we need to do is calculate the length and then we can calculate the area of the Shaded region so how can we do this well we need to know that the diagonals of a rhombus they meet at right angles and also they bisect each other so these two sides are congruent and those two segments are congruent so therefore D1 is eight that means that these two sides they're four let's call this a b c d e so now we need to calculate segment a e and EC or just easy so notice that we have a right triangle so we could find a Miss inside this side is 4 this side is 5. so let's calculate a c squared is equal to a squared plus b squared so C is five and B is 4. so 5 squared is 25 4 squared is 16. 25 minus 16 is 9. and so if we take the square root of both sides a is 3. so now we have everything that we need in order to calculate the area so if e c is 3 a e is also 3 which means that the left of the triangle I mean the left of the rectangle is six three plus three is six so the area is going to be one half times six times eight half of six is three three times eight is twenty-four so the area is 24 square units now let's say if we have a circle and let's say there's a triangle around the circle and let's say the radius of the circle is 20. what is the area of the Shaded region now last time when we had a circle and a rectangle the rectangle was the square because the circle is even all around so this type of triangle around a circle has to be an equilateral triangle and notice that if you draw a line where it touches the triangle and a circle that's r and notice it's the same everywhere for all three points where the circle meets the triangle so you need to realize that this is an equilateral triangle and so if we can calculate s we can calculate the area of the equilateral triangle so R is 20. let's turn this into a right triangle so this side is 20. how can we calculate this part because if this is s then this part is half of s all we need is an angle if we could find one of these two angles then we can do it so the question is how can we find one of those angles so notice that if we draw a line between a center and the vertex of the triangle we can draw three of such lines and the angle of a full circle is 360. so if you take 360 and divide it by 3 that will give you 120 which is this angle and that's also this angle so therefore this angle here must be half of 120 which is 60. so if this angle is 60 the other one is 30. so we have a 30 60 90 right triangle now across the 30 let's say this is 2. decide across the 30 is going to be half of the hypotenuse so that's going to be 1. and the side across the 60 is whatever this value is times the square root of 3. now we have this side across the 20 I mean across the 30 which is 20. so the hypotenuse is twice that value it's 40. and so the side across the 60 is whatever this is times the square root of 3. so it's 20 square root 3. and so that's s over 2. so if that's half of s itself has to be twice the value so s is 40 square root 3. and so now we have enough information to calculate the area of the Shaded region the area of the Shader region is going to be the area of the triangle minus the area of the circle the area of an equilateral triangle is the square root of three divided by 4 times s squared and the area of a circle is pi r squared so s is 40 square root 3. and r is 20 that was given to us at the beginning now 40 squared that's 40 times 40 that's 1600. and the square root of 3 squared is just 3. 20 times 20 is 400. now 1600 divided by 4 that's 400 and 400 times 3 is 1200. so the area is going to be 1200 square root 3 minus 400 pi so this is the exact answer for the area of the Shaded region