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Surface Area and Volume of Combination of Solids

We study about solids in a branch of geometry, called solid geometry. A solid is a three dimensional figure; therefore, solid geometry is also termed as 3D geometry. Few main solid geometrical figures are cube, cuboid, sphere, hemisphere, prism, pyramid, cone etc. In solid geometry, we also study about the surface areas (the area of the outer surface of the solid) and volumes (the amount of space occupied in the solid) of solids and combinations of solids.
A combination of solid is said to be a solid figure formed by combing two or more separate solids. Such as - A tent may be made of a hollow cone placed on top of a hollow cylinder or an ice-cream cone is the combination of a cone and a hemisphere on top. We should be careful while dealing with the calculations of surface areas and volumes of these combinations of solids. In this article, we are going to learn about surface areas and volumes of combinations of solid figures.

Related Calculators
Calculate Surface Area Calculate Surface Area of a Circle
Calculator for Surface Area of a Cylinder Cone Surface Area Calculator
 

Formulae

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Surface Areas of Solids
The formulae of most-often used solid figures are listed below:

Cube

Lateral Surface Area (Area of 4 walls) = 4 (side)$^{2}$

Total Surface Area = 6 (side)$^{2}$

Cuboid

Lateral Surface Area (Area of 4 walls) = 2 (l + b) x h

Total Surface Area = 2(lb + bh + hl)

Where, l, b and h stand for length, breadth and height.

Cylinder
Curved Surface Area = 2 $\pi$ r h

Total Surface Area = 2 $\pi$ r (h + r)

Where, r and h represent radius and height.

Cone
Curved Surface Area = $\pi$ r l

Total Surface Area = $\pi$ r (l + r)

Where, r and h represent radius and slant height.

l = $\sqrt{r^{2}+h^{2}}$

Sphere

Total Surface Area = Curved Surface area = $4 \pi r^{2}$

Hemisphere

Curved Surface area = $2 \pi r^{2}$

Total Surface Area = $3 \pi r^{2}$

Pyramid
Lateral Surface Area = $\frac{1}{2}$ x perimeter of base x slant height

Total Surface Area = Area of base + Lateral surface area

= Area of base + $\frac{1}{2}$ x perimeter of base x slant height

Prism
Lateral Surface Area = perimeter of base x slant height

Total Surface Area = 2 x Area of base + Lateral surface area

= (2 x Area of base) + (perimeter of base x slant height)

Volumes of Solids
The formulae of most-common solid figures are given below:

Cube

Volume = (side)$^{3}$

Cuboid

Volume = l b h

Where, l, b and h are length, breadth and height of the cuboid.

Cylinder
Volume = $\pi r^{2} h$

Where, r and h are radius and height of the cylinder.

Cone
Volume = $\frac{1}{3}$ $\pi r^{2} h$

Where, r and h are radius and height of the cone.

Sphere

Volume = $\frac{4}{3}$ $\pi r^{3}$

Hemisphere

Volume = $\frac{2}{3}$ $\pi r^{3}$

Pyramid

Volume = $\frac{1}{3}$ x area of base x height

Prism
Volume = area of base x height

How to Find

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We need to be very careful while finding the surface areas and volumes of the combinations of solids. In order to calculate the surface area of any combination of solid, one needs to look for the exposed area of the combination of solids.
Then we should use the suitable formula for each solid, find out the exposed surface area of each solid in the combination and calculate their sum.

In order to calculate the volume of combination of solids, one needs to find the volume of each solid by using separate formulae and then sum them up.
For Example: If we are supposed to find the outer surface area and volume of the tent which is made up of a cone over a cylinder, then we will have to first determine the dimensions of both cone and cylinder.

Then, to find total surface area of tent, we have the add the curved surface areas of cone and cylinder. Also, for finding the volume of the tent, one needs to calculate the volumes of cone and cylinder separately and then add them together.

Examples

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Few examples of surface areas and volumes of are as follows:

Example 1:
A cone is surmounted on a same diameter cylinder to make the shape of a tent. If the diameter of cylinder be 40 m and heights of conical and cylindrical parts are 2.1 meter and 4.2 meter. Calculate the capacity of the tent.

Solution:

The diagram of the tent is given below.
Cylinder
For cylinder

r = 20 m, h = 4.2 m

Volume of the cylinder = $\pi r^{2} h$

= $\frac{22}{7}$ $\times 20^{2} \times 4.2$

For cone

r = 20 m, h = 2.1 m

Volume of the cone = $\frac{1}{3}$ $\pi r^{2} h$

= $\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ 20$^{2}$ $\times$ 2.1

Volume of tent = $\frac{22}{7}$ $\times$ 20$^{2}$ $\times$ 4.2 + $\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times 20^{2} \times 2.1$

= $\frac{22}{7}$ $\times 400 \times 4.2 + $\frac{1}{3}$ $\times$ $\frac{22}{7}$ $\times$ 400 $\times$ 0.7

= $\frac{22}{7}$ $\times$ 400 $\times$ (4.2 + 0.7)

= $\frac{22}{7}$ $\times 400 \times 4.9$

= $22 \times 400 \times 0.7$

= 6160

Volume of the tent is 6160 cubic meter.

Example 2: In the figure below, a toy is shown. If the radius and height of the conical portion is 3 cm and 4 cm. Find the surface area of the toy. (use $\pi$ = 3.14)
Cone

Solution: For the cone

r = 3 cm

h = 4 cm

l = $\sqrt{3^{2}+4^{2}}$ = 5 cm

Curved surface area of cone = $\pi$ r l

= $\pi$ 3 x 5 = 15 $\pi$

For hemisphere

r = 3 cm

Curved surface area = $2 \pi r^{2}$

= $2 \pi \times 3^{2}$ = 18 $\pi$

Surface area of the toy = 15 $\pi$ + 18 $\pi$

= 33 $\pi$

= 33 x 3.14

= 103.62

Thus surface area of the toy is 103.62 sq cm

Example 3: In the following diagram attached, two hemispheres are attached each on both circular ends of a cylinder of length 50 cm. The radius is given as 14 cm. Find the volume of the solid so obtained.
Two Hemisphere
Solution: For cylinder

r = 14 cm, h = 50 cm

Volume = $\pi$ r$^{2}$ h

= $\frac{22}{7}$ x 14 x 14 x 50

For hemisphere

r = 14 cm

Volume of two hemispheres = 2 x $\frac{2}{3}$ $\pi r^{3}$

= $\frac{4}{3}$ x $\frac{22}{7}$ x 14 x 14 x 14

Volume of the toy = $\frac{22}{7}$ x 14 x 14 x 50 + $\frac{4}{3}$ x $\frac{22}{7}$ x 14 x 14 x 14

= $\frac{22}{7}$ x 14 x 14 (50 + $\frac{56}{3}$)

= 616 (50 + $\frac{56}{3}$)

= 42298.67

Therefore the volume of the toy is 42298.67 cubic cm.
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