 Hello and welcome to the session. In this session we will discuss about the length of the diagonal of a cuboid and a cube. First let's discuss length of the diagonal of a cuboid. Consider this cuboid pq rs p dash q dash r dash s dash. We suppose length of the cuboid b equal to a then breadth of the cuboid b equal to b height of the cuboid b equal to c. So this is a that is the length of the cuboid, breadth of the cuboid is b and height of the cuboid is c like this qs dash b one of the diagonals of the cuboid and we have drawn this qs as the diagonal of the rectangle pq rs. Consider this triangle pqs this triangle pqs is the right triangle where angle qps is of measure 90 degrees. So we can apply the Pythagoras theorem. So by applying the Pythagoras theorem in triangle pqs we have qs square that is the hypotenuse square is equal to the perpendicular square that is ps square plus the base square which is pq square. We have pqs a and ps as b so this means qs square is equal to b square plus a square or you can say qs square is equal to a square plus b square. Let this be result one. Now consider the triangle qss dash this is the right triangle that is triangle qss dash is the right triangle such that angle qss dash is equal to 90 degrees that is this triangle qss dash is right angled at s. Now since this is the right triangle so we can apply Pythagoras theorem to this triangle so by the Pythagoras theorem in the triangle qss dash we have the hypotenuse square which is qs dash square is equal to the base square which is qs square. So where we write qs square plus the perpendicular square the perpendicular in this triangle is ss dash so we write here ss dash square and in this figure we have ss dash is equal to c and from result one we have qs square is equal to a square plus b square. So here we have qs dash whole square is equal to a square plus b square plus c square that is ss dash is c so here we have c square and qs square is equal to a square plus b square. So this further means that qs dash is equal to square root of a square plus b square plus c square and this qs dash is one of the diagonals of the given cuboid. This means that length of the diagonal qs dash is equal to square root of a square plus b square plus c square and in general we say that length of the diagonal of a cuboid is equal to square root of length square plus b square plus i square. So this is how we can find the length of the diagonal of the cuboid given its length, width and height. Now next we will discuss length of the diagonal of a cube. We know that a cube is a special case of a cuboid in which all its edges are equal and each face of a cube is a square. So consider this cube in which all the edges are of measure a that is all the edges are equal that is the length, breadth and height of a cube are equal and we have taken here it to be a. So putting length as a, breadth as a and height as a in this formula for the length of the diagonal of a cuboid we can find out the length of the diagonal of a cube. So we can now say that length of the diagonal of a cube is equal to square root of length square that is a square plus breadth square which is a square plus the height square that is a square. And this is equal to square root of 3a square or you can say root 3 into a that is root 3 into length of the side of the cube which is a in this case. So we can say that length of the diagonal of a cube is equal to root 3 into height of the cube. So when we are given the length of the side of the cube we can easily find out the length of the diagonal of a cube. Let us consider an example in which we are given length of the cuboid as 3 centimeters, breadth of the cuboid as 4 centimeters and height of the cuboid as 5 centimeters. So length of the diagonal of the cuboid is equal to square root of length square that is 3 square plus breadth square which is 4 square plus height square that is 5 square. So further this is equal to square root of 9 plus 16 plus 25 that is equal to square root 15 which is further equal to 5 root 2 centimeters. That is for the given length breadth and height of the cuboid length of the diagonal of the cuboid is 5 root 2 centimeters. Give another example where we are given the length of the height of a cube as 4 centimeters. The length of the diagonal of the cube is given by root 3 into the length of the side of the cube that is 4 centimeters. So for root 3 centimeters is the length of the diagonal of the cube whose side is of measure 4 centimeters. So this completes the session. Hope you understood how to find the length of the diagonal of the cuboid and cube.