. Volume and Surface Area of Combined Solids Observe the shapes given (Fig. . ).
The shapes provided lead to the following definitions of ‘Combined Solid’. Fig. . h h Mensuration A combined solid is said to be a solid formed by combining two or more solids.
The concept of combined solids is useful in the fields like doll making, building construction, carpentry, etc. To calculate the surface area of the combined solid, we should only calculate the areas that are visible to our eyes. For example, if a cone is surmounted by a hemisphere, we need to just find out the C.S.A. of the hemisphere and C.S.A.
of the cone separately and add them together. Note that we are leaving the base area of both the cone and the hemisphere since both the bases are attached together and are not visible. But, the volume of the solid formed by joining two basic solids will be the sum of the volumes of the individual solids. Example .
A toy is in the shape of a cylinder surmounted by a hemisphere. The height of the toy is cm. Find the total surface area of the toy if its common diameter is cm. Solution Let r and h be the radius and height of the cylinder respectively.
Given that, diameter d = cm, radius r = cm Total height of the toy is cm Therefore, height of the cylindrical portion = = cm T.S.A. of the toy = C.S.A. of the cylinder + C.S.A. of the hemisphere +Base Area of the cylinder rh = p r h sq.
units × × × × = Therefore, T.S.A. of the toy is cm Example . A jewel box (Fig. .
) is in the shape of a cuboid of dimensions cm ´ cm ´ cm surmounted by a half part of a cylinder as shown in the figure. Find the volume of the box. Solution Let l, b and h be the length, breadth and height of the cuboid. Also let us take r and h be the radius and height of the cylinder.
cm cm cm Tent Ice Cream Capsule Tanjore Doll Fig. . Now, Volume of the box = Volume of the cuboid + (Volume of cylinder) × × l h r h p cu. units