Surface Area of a Combination of Solids

Surface area is the total area occupied by the surface of a three-dimensional object. It is the sum of the areas of all the surfaces of the object.

There are two main types of surface area (SA) in solid geometry:

• Total Surface Area (TSA): It is the area of all surfaces of a 3D object, including both curved and flat surfaces.
• Curved Surface Area (CSA): It is the area of only the curved part of a 3D object, excluding any flat surfaces.

Combination of Two Solids

When you have a combination of two solids, such as a cylinder with a cone on top, or a cube with a hemisphere on one of its faces, calculating the total surface area involves finding the surface areas of each individual solid and then summing them up.

Cone and Hemisphere

This figure can be simplified into a cone and a hemisphere. But note that while calculating the curved surface area, we do not consider the base of the hemisphere and cone, as it is not exposed outside. Hence, we only add the curved surface area of the solids.

∴ S.A. of figure = C.S.A of cone + C.S.A of hemisphere

S.A. of figure = πrl + 2πr²

S.A. of figure = πr(l+2r)

Cylinder and Cone

For the total surface area of this solid, we need to take into account the curved surface area of the cylinder, the area of the base of the cylinder and the curved surface area of the cone (not TSA as the base of cone is inside the solid)

∴ S.A. of figure = CSA of cylinder + base area of cylinder + CSA of cone

S.A. of figure = 2πrh + πr² + πrl

S.A. of figure = πr(2h + r + l)

Cylinder and Cube

Here, since there is an overlapped area of the top of the cylinder, here is how we calculate the surface area of this solid:

∴ S.A. of figure = (CSA of cylinder + area of base of cylinder + TSA of cube) - area of top of cylinder

S.A. of figure = (2πrh + πr² + 6a²) - πr²

S.A. of figure = 2πrh + 6a²

Cone and Cube

Here, since there is an overlapped area of the top of the cone, here is how we calculate the surface area of this solid:

∴ SA of solid = (CSA of cone + TSA of cube) - area of top of cone

SA of solid = ( πrl + 6a²) - πr²

SA of solid = πr(l + r) + 6a²

Cube and Hemisphere

Here, since there is an overlapped area of the hemisphere, here is how we calculate the surface area of this solid:

∴ SA of solid = (SA of cube + SA of hollow hemisphere) - base area of hemisphere

SA of solid = (6a² + 2πr²) - πr²

SA of solid = 6a² + πr²

Combination of Three Solids

Now that we have seen the combination of two solids, let us look at a more complex topic: combination of three solids.

Cylinder and Two Cones

Here, let us consider that the cones are identical and base radius of the cone and the cylinder is the same.

∴ SA of solid = CSA of cylinder + 2 x (CSA of cone)

SA of solid = 2πrh + 2 x (πrl)

SA of solid = 2πr(h + l)

Cylinder and Two Hemispheres

Here, let us consider that the hemispheres are identical and base radius of the hemispheres and the cylinder is the same.

∴ SA of solid = CSA of cylinder + 2 x (CSA of hemisphere)

SA of solid = 2πrh + 2 x (2πr²)

SA of solid = 2πrh + 4πr²

SA of solid = 2πr(h + 2r)

Application in Real-Life Examples

Knowing the surface area of combination of solids can be very helpful in everyday-life as these combinations can be objects we use in real life as well. Some practical applications include:

• While constructing buildings or structures, civil engineers and architects need to know the surface areas of the complex structures to estimate the amount of paint required, to optimize designs to reduce costs. Etc.

• In designing efficient applications of thermal heat transfers such as refrigerators, cooling systems and radiators, designers need to calculate the surface areas of the surfaces to maximize heat transfer efficiency.

• Medication dosage to patients is given in proportion to the surface area of the individual for its uniform distribution over the body.

These are just a few applications that show us the importance of knowing the concept of surface area and how it is deeply related to our everyday life.

Related Articles:

• Surface Area Formulas
• Surface Areas and Volumes
• Real Life Applications of Surface Area

Solved Problems on Surface Area of Combinations of Solids

Example 1: Find the total surface area of a cylindrical tin can with a hemispherical lid. The radius of the cylindrical part is 5 cm and the height is 12 cm.

Solution:

Surface area of the cylindrical part = 2πrℎ = 2 × π × 5 × 12 = 120π sq. cm

Surface area of the hemispherical lid = 2πr² = 2 × π × 5² = 50π sq. cm

Total surface area = Surface area of cylindrical part + Surface area of hemispherical lid

Total surface area = 120π + 50π= 170π sq. cm

Example 2: A cone is placed over a hemisphere such that their bases coincide. If the radius of the hemisphere is 6 cm and the height of the cone is 8 cm, find the total surface area of the combination.

Solution:

Surface area of the hemisphere = 2πr² = 2 × π × 6² = 72π sq. cm

Slant height (l) of the cone = √r²+ ℎ² = √6²+ 8² = √100 = 10 cm

Curved surface area of the cone = πrl = π × 6 × 10 = 60π sq. cm

Total surface area = Surface area of hemisphere + Curved surface area of cone

Total surface area = 72π + 60π = 132π sq. cm

Example 3: A solid iron pole consists of a cylinder of height 220 cm and base radius 9 cm, surmounted by another cylinder of height 60 cm and radius 7 cm. Find the total surface area of the pole.

Solution:

Surface area of the first cylinder = 2πrh = 2 × π × 9 × 220 = 3960π sq. cm

Surface area of the second cylinder = 2πrℎ = 2 × π × 7 × 60 = 840π sq. cm

Area of the circular end of the first cylinder = πr²= π × 9²= 81π sq. cm

Area of the circular end of the second cylinder = ??²=?×7²=49? sq. cm

Total surface area = Surface area of first cylinder + Surface area of second cylinder + Area of circular ends

Total surface area = 3960π + 840π + 81π + 49π = 4930π sq. cm