Let us find out the surface areas of all these objects !

Cuboid Cube

Cylinder

Cone

Sphere

Surface area of a cuboidSurface area of a cuboid

A bundle of many sheets of A bundle of many sheets of paper makes up a cuboid.paper makes up a cuboid.

We have used six We have used six rectangular pieces to cover rectangular pieces to cover the complete outer surface the complete outer surface of the cuboid.of the cuboid.

Area of 6 rectangles = Area of 6 rectangles = 2(lxb) + 2(bxh) + 2(lxh)2(lxb) + 2(bxh) + 2(lxh)

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

Surface area of a cubeSurface area of a cube

A cuboid, whose length, A cuboid, whose length, breadth and height are all breadth and height are all equal, it is called a Cube. If equal, it is called a Cube. If each edge of the cube is a, each edge of the cube is a, then the surface area of this then the surface area of this cube would be:cube would be:

2(axa+axa+axa),i.e.,6a2(axa+axa+axa),i.e.,6a², giving ², giving us – us –

Surface area of a cube =6a²Surface area of a cube =6a²

where a is the edge of the cube.where a is the edge of the cube.

Surface area of a right circular Surface area of a right circular cylindercylinder

Take a number of Take a number of circular sheets of paper circular sheets of paper and stack them up, if the and stack them up, if the stack is kept vertically stack is kept vertically up, we get a Right up, we get a Right circular cylinder, since circular cylinder, since it has been kept at right it has been kept at right angles to the base, and angles to the base, and the base is circular.the base is circular.

The area of the rectangular sheet The area of the rectangular sheet gives us the curved surface area gives us the curved surface area of the cylinder. The length of the of the cylinder. The length of the sheet is equal to the sheet is equal to the circumference of the circular base circumference of the circular base which is equal to 2which is equal to 2חחr.r.

So, curved surface of the cylinder So, curved surface of the cylinder = area of the rectangular sheet = = area of the rectangular sheet = length x breadthlength x breadth

= perimeter of the base of the = perimeter of the base of the cylinder x hcylinder x h

r x hr x hחח 2 = 2 = Therefore, Curved surface area of a Therefore, Curved surface area of a

cylinder = cylinder = 22חחrh.rh.

If the top and the bottom of If the top and the bottom of the cylinder are also to be the cylinder are also to be covered, then we need two covered, then we need two circles to do that, each of circles to do that, each of radius r, and thus having an radius r, and thus having an area of area of חחr² each, giving us r² each, giving us the total surface area as the total surface area as

. r²(r+h) .r²(r+h)חח r² = 2 r² = 2חח rh + 2 rh + 2חח 2 2So, Total surface area of a So, Total surface area of a

cylinder =2 cylinder =2 חחr (r +h)r (r +h)Where h is the height of the Where h is the height of the

cylinder and r is it’s radius.cylinder and r is it’s radius.

Surface area of a right circular coneSurface area of a right circular cone The area of each triangle = The area of each triangle = ½ x base of ½ x base of

each triangle x l.each triangle x l.So, area of the entire piece of paperSo, area of the entire piece of paper = sum of the areas of all the triangles= sum of the areas of all the triangles = ½ b1= ½ b1l+ l+ ½b2½b2l+ l+ ½b3½b3l+…l+… = = ½½l(b1+b2+b3+…)l(b1+b2+b3+…) = = ½ x ½ x l l x length of entire x length of entire

curved boundary of curved boundary of the fig 3the fig 3

(as b1+b2+b3+… makes up the curved portion (as b1+b2+b3+… makes up the curved portion of the fig)of the fig)

But the curved portion of the figure makes up But the curved portion of the figure makes up the perimeter of the base of the cone and the perimeter of the base of the cone and the circumference of the base of the cone = the circumference of the base of the cone = .r, where r is the base radius of the cone.r, where r is the base radius of the coneחח 2 2

So, So, Curved surface area of a cone Curved surface area of a cone = ½ x 1 x 2 = ½ x 1 x 2 חחr = r = חחrrl. l. Where r is it’s base radius and l. it’s slant Where r is it’s base radius and l. it’s slant

height. height. TSA of a cone = TSA of a cone = חחr r l. + l. + חחr² = r² = חחr (l+r)r (l+r)

Surface area of a sphereSurface area of a sphere The string which had completely The string which had completely

covered the surface area of the covered the surface area of the sphere, has been used to sphere, has been used to completely fill the regions of four completely fill the regions of four circles, all of the same radius as circles, all of the same radius as of the sphere.of the sphere.

This suggests that the surface area This suggests that the surface area of a sphere of radius rof a sphere of radius r

= 4 times the area of a circle = 4 times the area of a circle of radius r =4 x (of radius r =4 x ( חחr²)r²)

So, So, Surface area of a sphere =4 Surface area of a sphere =4 חחr².r².Where r is the radius of the sphere.Where r is the radius of the sphere.

EvaluationEvaluation A cubic box has each edge 10 cm and another cuboidal box is 12.5 cm A cubic box has each edge 10 cm and another cuboidal box is 12.5 cm

long, 10 cm wide and 8 cm high.long, 10 cm wide and 8 cm high.

Which box has the grater surface area and by how much?Which box has the grater surface area and by how much? A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the cost

of painting the curved surface of the pillar at the rate of Rs 12.50 per mof painting the curved surface of the pillar at the rate of Rs 12.50 per m² .² . Diameter of the base of a ice cream cone is 10.5 cm and its slant height is Diameter of the base of a ice cream cone is 10.5 cm and its slant height is

10 cm. Find its curved surface area.10 cm. Find its curved surface area. Find the surface area of a sphere of radius: 14 cm.Find the surface area of a sphere of radius: 14 cm.

Find the radius of a sphere whose surface area is 154 cmFind the radius of a sphere whose surface area is 154 cm².².