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 m3. 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
地点: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- 2 a b-2a h - 2 a。
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= a3
(ii)Surface area of cube = 2 (aa + aa + aa)
= 2(a2+ a2+ a2)
= 2(3a2) = 6a2= 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
= (pr2) × h
=pr2h m3or cm3
(2)Curved surface = (Perimeter of base) × height
= (2 ×pr) × h
= 2prh m2or cm2
(3)Total surface area = Area of circular ends + Curved surface area
= 2pr2+ 2prh
= 2pr (r + h) m2or cm2
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 =pR2h -pr2h
=ph (R2- r)2
(2)Curved surface = 2pRh + 2prh
= 2ph (R + r)
(3)Total surface area = 2ph (R + r)
+ 2p(R2- r2)
= 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 bya.
(1)Volume of Cone = 1/3(Area or base) × height
= 1/3(pier2) × h = 1/3pier2h cubic units.
(2)Curved Surface Area =pierlsq. units.
(3)Total Surface Area = (pierL +pier2) 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 and2= h2+ (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
=pR2+pr2+p
(R + r) =pie[R2+ r2+ l(R + r)] sq. units.
Where l= slant height, r = radius of circular base, h = height.
Where l2= h2+ r2
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 = 4/3pier3cubic units.
(2)Surface area of sphere = 4piel2
Hemisphere:
(1)Volume of hemisphere =
(2)Curved area:Surface area of hemisphere = 2pr2
(3)Surface (total surface area) of solid hemisphere = 2pr2+pr2= 3pr2
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 =区域的e 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
o =表面积+区域f the base sq.
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