Can you find area of the Green shaded Square? | (Quarter Circle) | #math #maths | #geometry

Añadido: 2026-05-08

[00:00:00]Welcome to pre-math. In this video, we have got this green shaded square BDEF and the purple shaded right triangle CDE put together as you can see in this given diagram and both of these are fully confined inside this quarter circle with the center B such that this point E is on this quarter circle and furthermore the area of this purple shaded right triangle has been given to us as 5 cm² and now our task is to calculate the area of this green shaded square BDEF.

[00:00:50]Please don't forget to give a thumbs up and subscribe and please keep in mind that this figure may not be 100% true to the scale.

[00:00:58]Let's go ahead and get started and here's our very first step. Let's focus once again on this purple shaded right triangle and bear in mind that this angle has got to be our 90° angle since we are dealing with this square and we know the area of this triangle has been given to us as 5 cm² and now let's recall the area of a triangle formula. Area is always equal to a half times base times the height of the triangle and in our case the base of this purple shaded triangle is this segment CD length whereas the height of this triangle is going to be this DE length. So therefore now our task is to find the value of this segment CD length and this side DE length. And now we are going to focus on this green shaded square. Since we are interested in calculating the area of this green shaded square I'm going to label this side length of the square as lowercase A then this side length is going to be lowercase A, lowercase A and lowercase A across the board and now let's recall the area of a square formula. Area is always equal to S² where S represents the side length of the square and in our case the side length of this green shaded square is lowercase A. So therefore our this green shaded square area is going to be equal to A square. And now our task is to find the value of A square. And now in this next step I'm going to connect this point E with the center B as you can see in this next step and now we can see this BE is the radius of this quarter circle. I'm going to label this radius as lowercase R. And now our task is to find the value of this radius R.

[00:03:04]And now we are going to focus once again on this right triangle BDE and we are going to apply the Pythagorean theorem on this triangle and here's our Pythagorean theorem A² + B² = C². In our case our hypotenuse is radius R whereas our two other legs are lowercase A and lowercase A as well. Let's go ahead and fill in the blanks in this Pythagorean formula. We got A² + likewise A² = R² and now I'm going to undo this square by taking square root on both sides of this equation. So therefore our radius lowercase R value is going to be equal to A times the square root of 2. 2.

[00:03:54]So thus our this radius lowercase R value is going to be A times the square root of 2. And now let's make an observation. We can see this BC side length represents the radius lowercase R as well and our radius lowercase R value is A times square root of 2. So therefore this whole base BC length is going to be simply A times the square root of 2. And we know this segment BD length is lowercase A. So therefore this remaining segment CD length has got to be A times square root of 2 minus A. So I can write A times square root of 2 minus A. And now we are going to focus on this purple shaded right triangle CDE once again and now let's recall once again the area of a triangle formula. Area equals to a half times base times the height of the triangle and in our case the base of this triangle is A times square root of 2 minus A. The height is lowercase A and the area of this triangle is 5 cm². So therefore let's go ahead and fill in the blanks in this formula. So we got 5 as our area equals to a half times our base A times the height is A times square root of 2 minus lowercase A. And now let's make an observation inside this parentheses A is in common so I can write A times square root of 2 minus 1 as you can see in this next step. And now I'm going to multiply out A times A is going to give us A². So therefore we could write 5 = A² / 2 times 2 square root of 2 minus 1 and now I'm going to multiply by 2 on both sides on the left hand side and on the right hand side as well. And here we can see 2 and 2 is gone and 5 times 2 is going to give us 10. So I can write A² times square root of 2 minus 1 = 10.

[00:06:31]And now I'm going to divide both sides by square root of 2 minus 1 to isolate A². So therefore we can see this is gone so A² value is going to be equal to 10 / the square root of 2 minus 1. And now we are going to rationalize this denominator by multiplying by its conjugate the square root of 2 + 1 and at the same time I'm going to divide by the square root of 2 + 1 as well. And now I'm going to multiply out these numerators and at the very same time I'm going to multiply out these denominators as well. And now let's recall this famous identity A minus B times A plus B is going to give us A² minus B² and we are going to apply this identity on this denominator part. So therefore this denominator could be written as the square root of 2 whole square minus 1 square. So therefore that's going to give us 2 minus 1 equals to 1. So thus this denominator part has been simplified to 1 as you can see in this next step. So thus we are ended up with A² = 10 times square root of 2 + 1. And now let's recall once again we know the area of this green shaded square is A² and our A² value is 10 times square root of 2 + 1. So therefore we conclude that the area of this green shaded square is going to be 10 times the square root of 2 + 1 cm².

[00:08:23]So thus after all the calculations and manipulations the area of this green shaded square turns out to be 10 times square root of 2 + 1 cm² and that's going to be approximately equal to 24.14 cm² as well. And that's our final answer. Thanks for watching and please don't forget to subscribe to my channel for more exciting videos. Bye.