The concept of surface area and volume for Class 10 is provided here. In this article, we are going to discuss the surface area and volume for different solid shapes such as the cube, cuboid, cone, cylinder, and so on. The surface area can be generally classified into Lateral Surface Area (LSA), Total Surface Area (TSA), and Curved Surface Area (CSA). Here, let us discuss the surface area formulas and volume formulas for different three-dimensional shapes in detail. In this chapter, the combination of different solid shapes can be studied. Also, the procedure to find the volume and its surface area in detail.

Surface Area and Volume of Cuboid

A cuboid is the region covered by its six rectangular faces. The surface area of a cuboid is equal to the sum of the areas of its six rectangular faces.

Surface area of the cuboid

Consider a cuboid whose dimensions are $l\times b\times \mathrm{h,}$ respectively.

The total surface area of the cuboid (TSA) = Sum of the areas of all its six faces

TSA (cuboid) = $2(l\times b)+2(b\times h)+2(l\times h)=2(lb+bh+lh)$

Lateral surface area (LSA) is the area of all the sides apart from the top and bottom faces.

The lateral surface area of the cuboid = Area of face AEHD + Area of face BFGC + Area of face ABFE + Area of face DHGC

LSA (cuboid) = $2(b\times h)+2(l\times h)=2h(l+b)$

Length of diagonal of a cuboid =√(l${}^{}$+ b${}^{}$+ h${}^{}$)

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Volume of a Cuboid

The volume of a cuboid is the space occupied within its six rectangular faces.

Volume of a cuboid $=\left(basearea\right)\times height=\left(lb\right)h=lbh$

Surface Area and Volume of Cube

A cube is a three-dimensional solid, that has six square faces, twelve edges and eight vertices.

Surface Area of Cube

For a cube, length = breadth = height

TSA (cube) $=2\times \left(3l^2\right)=6l^2$
Similarly, the Lateral surface area of cube $=2(l\times l+l\times l)=4l^2$
Note: Diagonal of a cube $=\surd 3l$

Volume of a Cube

Volume of a cube = $basearea\times height$
Since all dimensions of a cube are identical, volume = $l^3$
Where l is the length of the edge of the cube.

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Surface Area and Volume of Cylinder

A cylinder is a solid shape that has two circular bases, connected with each other, through a lateral surface. Thus, there are three faces, two circular and one lateral, of a cylinder. Based on these dimensions, we can find the surface area and volume of a cylinder.

Surface Area of Cylinder

Take a cylinder of base radius r and height h units. The curved surface of this cylinder, if opened along the diameter (d = 2r) of the circular base can be transformed into a rectangle of length $2\pi r$ and height h units. Thus,

CSA of a cylinder of base radius r and height $h=2\pi \times r\times h$
TSA  of a cylinder of base radius r and height $h=2\pi \times r\times h$ + area of two circular bases
$=2\pi \times r\times h+2\pi r^2$
$=2\pi r(h+r)$

Volume of a Cylinder

Volume of a cylinder = Base area $\times$height = $\left(\pi r^2\right)\times h=\pi r^{2}h$

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Surface Area and Volume of Right Circular Cone

A cone is a 3d shape that has one circular base and narrows smoothly from base to a point, called vertex.

Surface area of cone

Consider a right circular cone with slant length l, radius r and height h.

CSA of right circular cone $=\pi rl$
TSA = CSA + area of base $=\pi rl+\pi r^2=\pi r(l+r)$

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Volume of a Right Circular Cone

The volume of a Right circular cone is 1/3 times that of a cylinder of the same height and base.
In other words, 3 cones make a cylinder of the same height and base.
The volume of a Right circular cone $=$$\mathrm{(1/3)\pi }{r}^{2}h$
Where ‘r’ is the radius of the base and ‘h’ is the height of the cone.

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Surface Area and Volume of Sphere

A sphere is a solid that is round in shape and the points on its surface are at equidistant from the center.

Surface area of Sphere

For a sphere of radius r

Curved Surface Area (CSA) = Total Surface Area (TSA) = $4\pi r^2$

For More Information On Sphere And Hemisphere, Watch The Below Video.

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Volume of Sphere

The volume of a sphere of radius r = (4/3)$\pi r^3$

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Surface Area and Volume of Hemisphere

A hemisphere is shape that is half of the sphere and has one flat surface. The other side of the hemisphere is shaped as a circular bowl. See the figure below.

Surface Area of Hemisphere

We know that the CSA of a sphere  $=4\pi r^2$.

A hemisphere is half of a sphere.
$\therefore$ CSA of a hemisphere of radius r $=2\pi r^2$
Total Surface Area = curved surface area + area of the base circle
$\Rightarrow$TSA $=3\pi r^2$

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Volume of Hemisphere

The volume (V) of a hemisphere will be half of that of a sphere.
$\therefore$ The volume of the hemisphere of radius r $=$$\mathrm{(2/3)\pi }{r}^3$

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Surface area and Volume of Combination of Solids

Combination of solids explains the shapes formed when two different solids are combined together. Thus, the surface area and volume for such shapes will vary from the other basic solids.

For More Information On Combination of Solids, Watch The Below Video.

Surface Area of Combined Figures

Areas of complex figures can be broken down and analysed as simpler known shapes. By finding the areas of these known shapes, we can find out the required area of the unknown figure.
Example: 2 cubes each of volume 64 $c{m}^3$  are joined end to end. Find the surface area of the resulting cuboid.
Length of each cube $={\mathrm{64(1/3)}}^{}$$=4cm$
Since these cubes are joined adjacently,  they form a cuboid whose length l = 8 cm. But height and breadth will remain the same = 4 cm.

$\therefore$ The new surface area, TSA = 2(lb + bh + lh)

TSA = 2 (8 x 4 + 4 x 4 + 8 x 4)

= 2(32 + 16 + 32)

= 2 (80)

TSA = 160 cm²

Volume of Combined Solids

The volume of complex objects can be simplified by visualising them as a combination of shapes of known solids.
Example: A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 3 cm and the height of the cone is equal to 5 cm.
This can be visualised as follows :

V(solid) = V(Cone) + V(hemisphere)

V(solid) = (1/3)$\pi r^2$h + (2/3)$\pi r^3$

V(solid) = (1/3)$\mathrm{\pi (9)(5)}$+ (2/3)$\mathrm{\pi (27)}$

V(solid) = 33$\mathrm{\pi \; cm3}$

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Surface area and volume of Frustum of a cone

When a solid is cut by a plane, then another form of solids is formed. One such form of solid is the frustum of a cone, which is formed when a plane cuts a cone parallelly to the base of the cone. Let us discuss its surface area and volume here.

If a right circular cone is sliced by a plane parallel to its base, then the part with the two circular bases is called a Frustum.

Surface Area of a Frustum

CSA of frustum $=\pi (r_1+r_2)l$,  where $l= \surd [h2+(r2 – r1)2]$

TSA of the frustum is the CSA + the areas of the two circular faces = $\pi (r_1+r_2)l+\pi ({\mathrm{r12}}_{}+{\mathrm{r22}}^{})$

Volume of a Frustum

The volume of a frustum of a cone $=$$\mathrm{(1/3)\pi }h(r12 + r22+r_1{r}_2)$

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Solved Example

When a solid is converted into another solid of a different shape (by melting or casting), the volume remains constant.

Suppose a metallic sphere of radius 9 cm is melted and recast into the shape of a cylinder of radius 6 cm. Since the volume remains the same after a recast, the volume of the cylinder will be equal to the volume of the sphere.

The radius of the cylinder is known however the height is not known. Let h be the height of the cylinder.
$r_1$ and $r_2$ be the radius of the sphere and cylinder, respectively. Then,
V(sphere) = V(cylinder)
$\Rightarrow \mathrm{4/3\pi }{r}_{1}3 =\mathrm{\pi r22}h$
$\Rightarrow 4/3\pi \left(9^3\right)=\pi \left(6^2\right)h$                   (On substituting the values)
$\Rightarrow h=27cm$