**Surface Area and Volumes**

By – XYZ

Subject – Maths, Sub. Teacher - Ms. XYZ

XYZ School,

XYZ

**Surface Area & Volume Defination**

Surface area is the total area of the faces and curved
surface of a solid figure.

Volume is
the quantity of three-dimensional space enclosed by some
closed boundary.

**Content**

Cuboid

Cube

Cylinder

Right Circular Hollow Cylinder

Right Circular Cone

Sphere

Spherical Shell

Hemispherical Shell

Frustum of a Right Circular Cone

**Cuboid **

A closed figure whose faces are rectangles, is called a
cuboid.

Let l, b and h are respectively the length, breadth and
height of a cuboid, then.

Volume = lbh cubic units

Total Surface Area of a Cuboid = 2(lb + bh + hl) sq. units.

Lateral Surface Area = 2(l + b)h sq. units.

Diagonal = √l2 + b2 + h2 units.

**Cube**

When all sides (edges) of a cuboid are equal in length, it
is called a cube.

If length of each egde of a cube is a units, then

Volume = a3 cubic units.

Total Surface Area = 6a2 sq. units.

Lateral Surface Area = 4a2 sq. units.

Diagonal = √3 a units.

**Cylinder**

A right circular cylinder is a solid generated by the
revolution of a rectangle about one of its sides.

Let r be the radius and h be the height of a cylinder. Then,

Volume = πrh

= Area of
base x height

Curved Surface Area =
2πrh

=
Perimeter of the base x height

Total Surface Area =
Curved Surface Area + Area of both circular ends

=
2πrh + 2πr²

=
2πr (h +r)

Area of each end =
πr²

**Right Circular Hollow Cylinder**

Let r and R be the internal and exterior radii of a hollow
cylinder of height h, Then,

Area of each end =
πR²-πr²= π(R²- r²)

Curved Surface Area =
Inner Curved Surface Area + Outer Curved Surface Area

=
2πrh + 2πRh

=
2πh (r+R)

Total Surface Area =
Curved Surface Area + Area of both ends

=
2πrh + 2πRh + πR²-πr² + πR²- πr²

= 2πh (r+R) + 2π(R²- r²)

= 2πh (r+R) + 2π(R+r)(R-r)

= 2π(r+R)(h+R-r)

Volume of material

= External Volume – Internal Volume

= πR²h-πr²h

= πh(R²-r²)

**Right Circular Cone**

If a right angle triangle is revolved about one of the sides
containing the right angle, the solid thus generated is called a right circular
cone.

If r, h and l be the radius, height and slant height of the
cone, then

l = √r² + h², r = √l² - h², h = √ l² - r²

Curved Surface Area =
πrl

Total Surface Area =
Curved Surface Area + Area of base

=
πrl + πr²

= πr(l +r)

Volume =
(1/3)πr²h

=
(1/3)(Area of base) x height

**Sphere**

It is the locus of a point in space, which moves such that
its distance from a fixed point is always constant.

If r be the radius of sphere, then

Surface Area = 4πr²

Volume = (4/3)πr³

For **Hemisphere**

Surface Area = 4πr²

Total Surface Area = 2πr² + πr² = 3πr²

Volume = (2/3)πr³

**Spherical Shell**

Let r and R be the inner and outer radii of a the sphere.

Volume of material = (2/3)πR³ - (2/3)πr³

C.S.A. (if edge ignored) : = 2πr² + 2πR²

C.S.A. (if edge included): 2πr² + 2πR² + πR² -πr²

**Frustum of Right Circular Cone**

Frustum of a cone is the solid obtained after removing the
upper portion of the cone by a cone parallel to its base. The lower portion
thus obtained is called frustum of the cone.

If the radii of bigger and smaller bases of the frustum are
‘R’ and ‘r’ slant height be ‘l’ then

Slant height (l) = √h² + (R-r)²

Curved Surface Area = π(R+r)l

Total Surface Area = C.S.A. + πR² + πr²

Volume = (1/3)πh(R² + r² +R.r)