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.

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Cube

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

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

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.

Curved Surface Area = 2 $\pi$ r h

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

Where, r and h represent radius and height.

Curved Surface Area = $\pi$ r l

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

Where, r and h represent radius and slant height.

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

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

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

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

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

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)

Cube

Volume = (side)$^{3}$

Volume = l b h

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

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

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

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

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

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

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

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

Volume = area of base x height

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.

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.

Example 1:

r = 20 m, h = 4.2 m

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

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

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.

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

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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