 Freaking as an Aston Professor in WIT, Sallapur. In this video lecture, we are going to learn about the development of the lateral surface of a square pyramid. At the end of this session, students will be able to draw the lateral surface of a regular pyramid like square, pentagonal, hexagonal pyramids. So this is a 3D diagram of a square pyramid. So basically it is a frustum. So here the apex is cut. If you see the first diagram, so here there is no apex. So it is having a base surface as a square, regular square. So and all these corners of the base square are connected at a vertex. So it is called a square pyramid. So it is having four triangular surfaces connecting to vertex to the base edges. So element of this square pyramid, we need to cut this pyramid at one of the inclined edge. So after that, so in the second figure you can see that, so it is cut at this inclined edge. So we can unroll all the surfaces so that we can kept it on a development plane. So in the third state it is opening, fourths diagram, fifths diagram and in the sixth diagram you can see that so all the triangular surfaces are kept on development plane and it is of top surface and the bottom. So like how we will get all the inclined surface or triangular surfaces kept on development plane, it is called as development of the pyramid. So we see, take one example on square pyramid. The question is that a square pyramid so which is having base 40 mm and axis height 65 mm which is rest on HP and all the edges of the base are equally inclined to VP. It is cut by a sectional plane or cutting plane which is perpendicular to VP that is vertical plane inclined at 45 degree to the horizontal plane and which is bisecting the axis, draw its sectional top view and true shape of the section also draw its development. So for this question here the square pyramid is resting on HP. So this is the answer for that. So if you see that, so how to draw the development, so this is called the development of that pyramid. To draw this according to the given condition of the question, here the base surface is resting on HP. So we need to draw one xy line so this the plane which is above the xy line is called as vertical plane and this plane is the horizontal plane. On the horizontal plane so according to the given condition the base surface is resting on HP horizontal plane. So from the top view we see that the square surface of the base of the pyramid will be seen with this true shape so that we need to draw a square in the top view with all of its edges like A, B, B, C, C, D and A, D. All these edges are equally inclined with VP. So from the top view we see that all the edges like A, D is 45 degree with xy line and similarly if this is 45 degree with xy line on the contrary C, D line is also at an angle of 45 degree with VP. So like how so these two lines are also will be 45 degree with xy line. So this satisfy the condition that so all the inclined edges are equally inclined with VP. Now so without cutting this is apex of the square pyramid which is having its full height and so this is the front view of the square pyramid. So these are top view so this O point is apex point in the top view here this from this point O2 this O dash this is the axis height. So from the top view we cannot see the height of the axis so it will appear as a point in the top view. So here we need to drop O dash point as a point so it is representing the axis of the pyramid. So from apex point O we need to collect all the base corners like O2A, O2B, O2C, O2D these are the inclined edges will be seen from the top view as these apparent lengths. Now I am going to draw this with a dark line so this surface is drawn because this pyramid is cut with this cutting plane section plane so which is perpendicular to VP. Since it is perpendicular to VP so from the front view we cannot see its true shape we see that section plane as a line so it is drawn here with a chain line. So now as per the given condition so we need to cut this object pyramid so which is passing at midpoint of the axis height so this is the midpoint of the axis so it should pass through this point and which is inclined with HP by 45 degree so this the inclination between this line and this line is 45 degree so we should draw one inclined 45 degree which is passing through the midpoint of the axis so that represent the cutting plane. So after that we need to remove this part this part of the pyramid that is apex side of the pyramid so this pyramid will remain after cutting or section. So for this pyramid we need to draw the development ok. Now so from this OA and OC are parallel to VP here so we need to project these points in the front view so that A will come here so from A to O we need to connect so this represents the true length of the inclined edge that is OA and OC it is also true length OC is also true length of the inclined edge of the pyramid. Now this B and D point to O point this represents the two inclined edges that is BO and DO so from B and D point since these two comes on same projection so these two points can be transferred on this x-y line so from this point we can draw the two more inclined edges but this is not the true length so this is the true length OA is a true length. Now without section for the full pyramid we can develop this surface as so we need to keep O point here first of all and we need to draw this OA line with the true length so because here I am taking here OA line because I am cutting this pyramid at OA inclined edge since it cut at OA inclined edge we need to draw this edge with the true length that is this length we need to measure with the compass and we need to cut here and from this point we need to cut one arc approximate length we need to cut at this radius OA radius now we need to divide this by R we can do that so A to D we need to measure with a compass or A to B and BC, CD and AD all are of equal length since it is a square so measure any one length and from this A point cut an arcs on this circle so from B to C and mark these points as A, B, C and D and again A because we need to have four triangular surfaces since it is a square means it is having four triangular surfaces we need to have four true shape of the triangular surfaces okay now name this as A, B, C, D and again A so this represents the development of the full pyramid without cut so if it is cut with this given condition so it is intersecting with this inclined edges at this point one point is coming on OA inclined edge and so next is BO inclined edge is here so the plane is intersecting with this point here name it as a two point similarly CO is intersecting here with cutting planes this is fourth and similarly DO is intersecting here so name this with 1, 2, 3, 4 and we need to and the next thing is that we need to transfer these heights from apex point or from base on this full development to draw the profile so I will transfer with O to 1 distance so it since it is on OA so measure with a compass and from this O point cut an arc on this so we get here and one more OA line is here so cut an arc on this so we get point number one and point number one so next is that next I will take that third point so it is on OC so measure this length with the help of compass and transfer this length on this OC edge next thing is that we need to transfer this 2 and 4 point we cannot directly measure this and we are supposed to transfer on here it is wrong we need to do that we need to transfer this 2 and 4 point first on this true length of the incline line since it is a true length we need to transfer this length on this true length that is OA now we measure the distance from this point to this point that is O to this point so this distance indicates that 2 and 4 at this distance and we need to measure this and transfer from this point on OB and OD so this is the correct method so after this we need to connect all this 1 2 3 4 1 point with a straight edge since all triangular surfaces are flat after cutting we get the profile as straight profile so after that so write this development with a dark line so this surface with a dark line indicates the development of the triangular sorry square pyramid to draw the true shape of this inclined surface or cut surface from the top view we can draw the auxiliary top view by seeing from this direction so for this inclined plane draw one XY X1 Y1 line which is parallel to this and from this you draw the projections and measure the distance of a point from XY line and transfer on this we get point here similarly from this point B and D point can be transferred here 2 and 4 point for that so C point can be transferred here so this can be transferred here so we get this shape as a true shape of the inclined cut surface and you can think that if the direction of the cutting plane if it is reverse if the cutting plane if I take like this what changes will appear on this development of the surface whether the profile is same or not or if I change the inclination by 45 degree to 30 degree whether the profile is same or not that you can think this is a reference I used thank you