Page 1 SURFACE AREAS AND VOLUMES Solid Figures: The objects which occupy space width, depth and height.) are called solids. The Example: Sphere, cone, cube, Surface Area: The sum of the areas of the plane figures making up the boundary of a solid figure is called its surface area Volume: The measure of part of space occupied by a solid is called its Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of i.e. Lateral area = Surface area Cuboid: A cuboid is a box-shaped solid object angles. All of its faces are rectangles. â€¢ Lateral Surface Area â€¢ Total Surface Area = â€¢ Volume of cuboid = Where, l = length b = breadth h = height Cube: If the faces of a rectangular parallelepiped be squares, then it is called â€¢ Lateral Surface Area = â€¢ Total Surface Area = â€¢ Volume of cube = Where, a = side. Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its sides is called a right circular cylinder. â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = r 2 p Where, r = radius of h = height SURFACE AREAS AND VOLUMES The objects which occupy space (i.e. they are 3 â€“ dimensional figures having are called solids. The solid figures are derived from the plane figures. , cylinder, rectangular prism, etc. The sum of the areas of the plane figures making up the boundary of a solid surface area. : The measure of part of space occupied by a solid is called its volume : Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of the top and the bottom. Lateral area = Surface area â€“ Area of two ends shaped solid object. It has six flat sides and all the angles are right ll of its faces are rectangles. Lateral Surface Area = h b l ) ( 2 + Total Surface Area = ) ( 2 hl bh lb + + Volume of cuboid = h b l × × b = breadth If the faces of a rectangular parallelepiped be squares, then it is called Lateral Surface Area = 2 4a Total Surface Area = 2 6a Volume of cube = 3 ) (a A solid obtained by revolving a rectangular lamina about one of its called a right circular cylinder. Curved Surface Area = rh p 2 Total Surface Area = ) ( 2 h r r + p h 2 , r = radius of circular base. dimensional figures having are derived from the plane figures. The sum of the areas of the plane figures making up the boundary of a solid volume. : Lateral surface area is the area of all the sides of an object where we It has six flat sides and all the angles are right If the faces of a rectangular parallelepiped be squares, then it is called cube. A solid obtained by revolving a rectangular lamina about one of its Page 2 SURFACE AREAS AND VOLUMES Solid Figures: The objects which occupy space width, depth and height.) are called solids. The Example: Sphere, cone, cube, Surface Area: The sum of the areas of the plane figures making up the boundary of a solid figure is called its surface area Volume: The measure of part of space occupied by a solid is called its Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of i.e. Lateral area = Surface area Cuboid: A cuboid is a box-shaped solid object angles. All of its faces are rectangles. â€¢ Lateral Surface Area â€¢ Total Surface Area = â€¢ Volume of cuboid = Where, l = length b = breadth h = height Cube: If the faces of a rectangular parallelepiped be squares, then it is called â€¢ Lateral Surface Area = â€¢ Total Surface Area = â€¢ Volume of cube = Where, a = side. Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its sides is called a right circular cylinder. â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = r 2 p Where, r = radius of h = height SURFACE AREAS AND VOLUMES The objects which occupy space (i.e. they are 3 â€“ dimensional figures having are called solids. The solid figures are derived from the plane figures. , cylinder, rectangular prism, etc. The sum of the areas of the plane figures making up the boundary of a solid surface area. : The measure of part of space occupied by a solid is called its volume : Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of the top and the bottom. Lateral area = Surface area â€“ Area of two ends shaped solid object. It has six flat sides and all the angles are right ll of its faces are rectangles. Lateral Surface Area = h b l ) ( 2 + Total Surface Area = ) ( 2 hl bh lb + + Volume of cuboid = h b l × × b = breadth If the faces of a rectangular parallelepiped be squares, then it is called Lateral Surface Area = 2 4a Total Surface Area = 2 6a Volume of cube = 3 ) (a A solid obtained by revolving a rectangular lamina about one of its called a right circular cylinder. Curved Surface Area = rh p 2 Total Surface Area = ) ( 2 h r r + p h 2 , r = radius of circular base. dimensional figures having are derived from the plane figures. The sum of the areas of the plane figures making up the boundary of a solid volume. : Lateral surface area is the area of all the sides of an object where we It has six flat sides and all the angles are right If the faces of a rectangular parallelepiped be squares, then it is called cube. A solid obtained by revolving a rectangular lamina about one of its SURFACE AREAS AND VOLUME Right Circular Cone: A right circular cone is a solid generated by the revolution of a right angled triangle about one of its side â€¢ Curved Surface Area â€¢ Total Surface Area = â€¢ Volume = 3 1 p Where, r = radius of base. h = height l = slant height Sphere: A sphere is a solid generated by the revolution of a semi â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = 3 4 p Where, r = radius Spherical shell: A spherical shell is a generalization of an annulus to three dimensions â€¢ Volume = 3 4 p Where, 1 r and 2 r are its â€¢ Total Surface Area Hemisphere: A plane passing through the centre each called a hemisphere. â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = 3 2 p Where, r = radius. SURFACE AREAS AND VOLUMES right circular cone is a solid generated by the revolution of a right angled triangle about one of its side containing the right angle as axis. Curved Surface Area = rl p = 2 2 h r r + p Total Surface Area = ) ( r l r + p h r 2 p , r = radius of base. l = slant height sphere is a solid generated by the revolution of a semi-circle about its diameter. Curved Surface Area = 2 4 r p Total Surface Area = 2 4 r p 3 r p , r = radius A spherical shell is a generalization of an annulus to three dimensions ) ( 3 2 3 1 r r - are its external and internal radii respectively. Total Surface Area = ) ( 4 2 2 2 1 r r + p A plane passing through the centre of a sphere cuts the sphere in two equal parts Curved Surface Area = 2 2 r p Total Surface Area = 2 3 r p 3 r p , r = radius. S right circular cone is a solid generated by the revolution of a right circle about its diameter. A spherical shell is a generalization of an annulus to three dimensions. respectively. cuts the sphere in two equal parts Page 3 SURFACE AREAS AND VOLUMES Solid Figures: The objects which occupy space width, depth and height.) are called solids. The Example: Sphere, cone, cube, Surface Area: The sum of the areas of the plane figures making up the boundary of a solid figure is called its surface area Volume: The measure of part of space occupied by a solid is called its Lateral Surface Area: Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of i.e. Lateral area = Surface area Cuboid: A cuboid is a box-shaped solid object angles. All of its faces are rectangles. â€¢ Lateral Surface Area â€¢ Total Surface Area = â€¢ Volume of cuboid = Where, l = length b = breadth h = height Cube: If the faces of a rectangular parallelepiped be squares, then it is called â€¢ Lateral Surface Area = â€¢ Total Surface Area = â€¢ Volume of cube = Where, a = side. Right Circular Cylinder: A solid obtained by revolving a rectangular lamina about one of its sides is called a right circular cylinder. â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = r 2 p Where, r = radius of h = height SURFACE AREAS AND VOLUMES The objects which occupy space (i.e. they are 3 â€“ dimensional figures having are called solids. The solid figures are derived from the plane figures. , cylinder, rectangular prism, etc. The sum of the areas of the plane figures making up the boundary of a solid surface area. : The measure of part of space occupied by a solid is called its volume : Lateral surface area is the area of all the sides of an object where we donâ€™t have to count the areas of the top and the bottom. Lateral area = Surface area â€“ Area of two ends shaped solid object. It has six flat sides and all the angles are right ll of its faces are rectangles. Lateral Surface Area = h b l ) ( 2 + Total Surface Area = ) ( 2 hl bh lb + + Volume of cuboid = h b l × × b = breadth If the faces of a rectangular parallelepiped be squares, then it is called Lateral Surface Area = 2 4a Total Surface Area = 2 6a Volume of cube = 3 ) (a A solid obtained by revolving a rectangular lamina about one of its called a right circular cylinder. Curved Surface Area = rh p 2 Total Surface Area = ) ( 2 h r r + p h 2 , r = radius of circular base. dimensional figures having are derived from the plane figures. The sum of the areas of the plane figures making up the boundary of a solid volume. : Lateral surface area is the area of all the sides of an object where we It has six flat sides and all the angles are right If the faces of a rectangular parallelepiped be squares, then it is called cube. A solid obtained by revolving a rectangular lamina about one of its SURFACE AREAS AND VOLUME Right Circular Cone: A right circular cone is a solid generated by the revolution of a right angled triangle about one of its side â€¢ Curved Surface Area â€¢ Total Surface Area = â€¢ Volume = 3 1 p Where, r = radius of base. h = height l = slant height Sphere: A sphere is a solid generated by the revolution of a semi â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = 3 4 p Where, r = radius Spherical shell: A spherical shell is a generalization of an annulus to three dimensions â€¢ Volume = 3 4 p Where, 1 r and 2 r are its â€¢ Total Surface Area Hemisphere: A plane passing through the centre each called a hemisphere. â€¢ Curved Surface Area = â€¢ Total Surface Area = â€¢ Volume = 3 2 p Where, r = radius. SURFACE AREAS AND VOLUMES right circular cone is a solid generated by the revolution of a right angled triangle about one of its side containing the right angle as axis. Curved Surface Area = rl p = 2 2 h r r + p Total Surface Area = ) ( r l r + p h r 2 p , r = radius of base. l = slant height sphere is a solid generated by the revolution of a semi-circle about its diameter. Curved Surface Area = 2 4 r p Total Surface Area = 2 4 r p 3 r p , r = radius A spherical shell is a generalization of an annulus to three dimensions ) ( 3 2 3 1 r r - are its external and internal radii respectively. Total Surface Area = ) ( 4 2 2 2 1 r r + p A plane passing through the centre of a sphere cuts the sphere in two equal parts Curved Surface Area = 2 2 r p Total Surface Area = 2 3 r p 3 r p , r = radius. S right circular cone is a solid generated by the revolution of a right circle about its diameter. A spherical shell is a generalization of an annulus to three dimensions. respectively. cuts the sphere in two equal parts SURFACE AREAS AND VOLUME 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 â€¢ Curved Surface Area Where, 1 2 (r h l + = â€¢ Total Surface Area = â€¢ Volume = 3 1 h p Where, h = vertical height l = slant height = 2 1 ,r r radii of the two bases. Surface Area of a Combination of solid: the curved surface areas of each of the individual parts. Example: TSA of new solid = CSA of one hemisphere + CSA of cylinder + Where, TSA= Total Surface Area and CSA=Curved Surface Area Volume of a Combination of Solids: two basics solids will actually be the sum of the volumes of the constituents. SURFACE AREAS AND VOLUME If a cone is cut by a plane parallel to the base of the cone, then the portion base is called the frustum of a cone. Curved Surface Area = ) ( 2 1 r r l + p 2 2 1 ) r - Total Surface Area = ) ( ) ( 2 2 2 1 2 1 r r r r l + + + p p ) ( 2 1 2 2 2 1 r r r r + + = vertical height = slant height radii of the two bases. Surface Area of a Combination of solid: The total surface area of the new solid is the sum of the curved surface areas of each of the individual parts. of new solid = CSA of one hemisphere + CSA of cylinder + CSA of other hemisphere Where, TSA= Total Surface Area CSA=Curved Surface Area Volume of a Combination of Solids: The volume of combination of the solid formed by joining will actually be the sum of the volumes of the constituents. SURFACE AREAS AND VOLUME If a cone is cut by a plane parallel to the base of the cone, then the portion The total surface area of the new solid is the sum of CSA of other hemisphere ombination of the solid formed by joining will actually be the sum of the volumes of the constituents.Read More

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