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# Area and volume of solid figure, math Assignment Help

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Geometry Mathematics - Area and volume of solid figure, math

**Area and volume of solid figure **

**Solids:** Bodies occupying space are called solids. A solid has three dimensions viz., length" breadth and thickness (or depth or height).

**Volume:** The space occupied by a solid body is called its volume. Volume is measured in cubic units. Its S.I. unit is m^{3}. i.e., the volume of a space region formed by a cube of m.** **

**Cuboid/rectangular parallelopiped: **

It is solid with six rectangular faces.

(i) Volume of cuboid = l × b × h

Where: l = length, b = breadth, h = height.

(ii) Surface area of cuboid = 2 (lb + bh + hl)

(iii) Surface area of 4 walls = 2(bh + hl) = 2(l + b) × h

(iv) Surface area of cuboid, in which top face is open

lb + 2 (bh + hl)

(v) Diagonal of faces of cuboid

(vi) Diagonal of cuboid =

(vii) Height of cuboid = volume/base area

(viii) Area of base = volume/height

**Note:**

Volume of material = Ext. volume - Int. volume.

**Closed cuboid:** If l, b, h are external dimension of closed cuboid of thickness 'a' then internal dimensions are : l - 2a, b-2a, h-2a.

**Open cuboid:** If the cuboid is an open cuboid then internal dimensions area l- 2a, b - 2a, h - a.

**Cube:** It is rectangular solid in which every face is square i.e.,

l = b = h = 'a' (say)

** **

(i) Volume of cube = (edge)^{3} = a^{3}

(ii) Surface area of cube = 2 (aa + aa + aa)

= 2(a^{2} + a^{2} + a^{2})

= 2(3a^{2}) = 6a^{2} = 6 (edge)^{2}

(iii) Diagonal of face of the cube = × a

(iv) Diagonal of cube = × a

(v) Edge of cube = (Volume)^{1/2}

**Right circular cylinder:** If a rectangle is revolved about its one side as its axis, the solid formed is called a Right circular cylinder.

**FORMULAE **

(1) Volume of cylinder = (Area of base) × height

= (pr^{2}) × h

= pr^{2} h m^{3} or cm^{3}

(2) Curved surface = (Perimeter of base) × height

= (2 × pr) × h

= 2prh m^{2} or cm^{2}

(3) Total surface area = Area of circular ends + Curved surface area

= 2pr^{2} + 2prh

= 2pr (r + h) m^{2} or cm^{2}

where r = radius of the circular base of cylinder

h = height of cylinder.

**Hollow cylinder:** Solids like iron pipes, rubber tubes etc. are in shape of hollow cylinder.

A solid bounded by two co-axial cylinders of the same height, is called a hollow cylinder.

**Cylinder with external and Internal radii:** Cylinder of height h and with external and internal radii Rand r respectively, we have.

(1) Volume of the material = pR^{2} h - pr^{2}h

= ph (R^{2} - r)^{2}

(2) Curved surface = 2pRh + 2prh

= 2ph (R + r)

(3) Total surface area = 2ph (R + r)

+ 2p (R^{2} - r^{2})

= 2p (R + r) (h + R - r).

** **

**Right circular cone:** If a right angled triangle is revolved about one of the sides containing a right angle, the solid thus formed is called a light circular cone.

**Slant height:** Slant height of a right circular cone is the distance of its vertex from any point on the circumference of the base. OL is the slant height of the cone.

**Semi vertical angle:** It is the angle between the height and the slant height and is usually denoted by a.

(1) Volume of Cone = 1/3 (Area or base) × height

= 1/3 (pier^{2}) × h = 1/3 pier^{2}h cubic units.

(2) Curved Surface Area = pierl sq. units.

(3) Total Surface Area = (pierL + pier^{2}) sq. units.

**Frustum of a cone:** If a cone is cut by a plane parallel to the base of the cone, then the portion between the plane and base is called the frustum of the cone.

**Lateral surface area of frustum of a right circular cone** = p(R + r) sq. units, where R, r be the radii of base and top of the frustum of a cone, h is the height of the frustum and ^{2} = h^{2} + (R -r)^{2}.

Volume of a frutrum of a cone =

**Total surface area of frustum of right circular cone **

= Area of base + Area of top + Lateral surface area

= pR^{2} + pr^{2} + p(R + r) = pie [R^{2} + r^{2} + l(R + r)] sq. units.

Where l = slant height, r = radius of circular base, h = height.

Where l^{2} = h^{2} + r^{2}

** **

**Sphere:** If a circle is revolved about its diameter, the solid thus formed is called a sphere.

** **

If r = radius of spare

(1) Volume of sphere = 4/3 pier^{3} cubic units.

(2) Surface area of sphere = 4 piel2

**Hemisphere:**

(1) Volume of hemisphere =

(2) **Curved area:** Surface area of hemisphere = 2pr^{2}

^{ }

(3) Surface (total surface area) of solid hemisphere = 2pr^{2} + pr^{2} = 3pr^{2}

A plane through the centre of the sphere cuts it into two equal parts. Each part is called hemisphere.

(4) **Spherical shell:** Volume of a spherical shell

**Prism:** A prism is a solid whose side faces are parallelograms and whose bases are equal and parallel rectilinear figures.

A prism is called a right prism, if the axis is perpendicular to the base.

Surface area of a right prism = Perimeter of the base × height sq. units.

Total surface area of a right prism = Lateral area + 2 (area of one box) sq. units.

Volume of right prism = Area of the base × height cu. units.

** **

**Pyramid****:** A polyhedron whose one face is a polygon and the other faces are triangles having a common vertex.

**Volume of a right pyramid** = 1/2 (area of the base) × height sq. units.

Surface area of a right pyramid = 1/2 (Perimeter of the base × slant height) sq. units.

Total surface area of a right pyramid

= Surface area + area of the base sq.

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**Area and volume of solid figure**

**Solids:**Bodies occupying space are called solids. A solid has three dimensions viz., length" breadth and thickness (or depth or height).

**Volume:**The space occupied by a solid body is called its volume. Volume is measured in cubic units. Its S.I. unit is m

^{3}. i.e., the volume of a space region formed by a cube of m.

**Cuboid/rectangular parallelopiped:**

**Note:**

**Closed cuboid:**If l, b, h are external dimension of closed cuboid of thickness 'a' then internal dimensions are : l - 2a, b-2a, h-2a.

**Open cuboid:**If the cuboid is an open cuboid then internal dimensions area l- 2a, b - 2a, h - a.

**Cube:**It is rectangular solid in which every face is square i.e.,

^{3}= a

^{3}

^{2}+ a

^{2}+ a

^{2})

^{2}) = 6a

^{2}= 6 (edge)

^{2}

^{1/2}

**Right circular cylinder:**If a rectangle is revolved about its one side as its axis, the solid formed is called a Right circular cylinder.

**FORMULAE**

^{2}) × h

^{2}h m

^{3}or cm

^{3}

^{2}or cm

^{2}

^{2}+ 2prh

^{2}or cm

^{2}

**Hollow cylinder:**Solids like iron pipes, rubber tubes etc. are in shape of hollow cylinder.

**Cylinder with external and Internal radii:**Cylinder of height h and with external and internal radii Rand r respectively, we have.

^{2}h - pr

^{2}h

^{2}- r)

^{2}

^{2}- r

^{2})

**Right circular cone:**If a right angled triangle is revolved about one of the sides containing a right angle, the solid thus formed is called a light circular cone.

**Slant height:**Slant height of a right circular cone is the distance of its vertex from any point on the circumference of the base. OL is the slant height of the cone.

**Semi vertical angle:**It is the angle between the height and the slant height and is usually denoted by a.

^{2}) × h = 1/3 pier

^{2}h cubic units.

^{2}) sq. units.

**Frustum of a cone:**If a cone is cut by a plane parallel to the base of the cone, then the portion between the plane and base is called the frustum of the cone.

**Lateral surface area of frustum of a right circular cone**= p(R + r) sq. units, where R, r be the radii of base and top of the frustum of a cone, h is the height of the frustum and

^{2}= h

^{2}+ (R -r)

^{2}.

**Total surface area of frustum of right circular cone**

^{2}+ pr

^{2}+ p(R + r) = pie [R

^{2}+ r

^{2}+ l(R + r)] sq. units.

^{2}= h

^{2}+ r

^{2}

**Sphere:**If a circle is revolved about its diameter, the solid thus formed is called a sphere.

^{3}cubic units.

**Hemisphere:**

**Curved area:**Surface area of hemisphere = 2pr

^{2}

^{ }

^{2}+ pr

^{2}= 3pr

^{2}

**Spherical shell:**Volume of a spherical shell

**Prism:**A prism is a solid whose side faces are parallelograms and whose bases are equal and parallel rectilinear figures.

**Pyramid**

**:**A polyhedron whose one face is a polygon and the other faces are triangles having a common vertex.

**Volume of a right pyramid**= 1/2 (area of the base) × height sq. units.

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