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
= Area of base x height
Curved Surface Area =
2πrh
= Perimeter of the base x height
= Perimeter of the base x height
Total Surface Area =
Curved Surface Area + Area of both circular ends
= 2πrh + 2πr²
= 2πr (h +r)
= 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)
= 2πrh + 2πRh
= 2πh (r+R)
Total Surface Area =
Curved Surface Area + Area of both ends
= 2πrh + 2πRh + πR²-πr² + πR²- πr²
= 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)
= 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)
= 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)
= πrl + πr²
= πr(l +r)
Volume =
(1/3)πr²h
= (1/3)(Area of base) x height
= (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)
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