CURVED SURFACE AREA AND TOTAL SURFACE AREA OF SPHERE AND HEMISPHERE

Curved Surface Area and Total Surface Area of Sphere and Hemisphere :

Here we are going to see, some example problems of finding curved surface area and total surface area of sphere.

Formula for Curved Surface Area and Total Surface Area of Hemisphere

Curved surface area

= 2Πr^{2}

Hollow hemisphere

= 2Π(R^{2} + r^{2}) sq. units

Total surface area of

= 3Πr^{2}

Hollow hemisphere

= Π(3R^{2} + r^{2}) sq. units

Curved Surface Area and Total Surface Area of hemisphere - Examples

Question 1 :

The radius of a sphere increases by 25%. Find the percentage increase in its surface area.

Solution :

Radius of hemisphere = r

Increased percentage of radius = 125% of r

= 125r/100

= 5r/4

Surface area of hemisphere = 2Πr^{2}

Surface area of increased hemisphere = 2Π(5r/4)^{2}

= 25Πr^{2}/8

= [(25Πr^{2}/8) - (2Πr^{2})]/(2Πr^{2})

= 9Πr^{2}/8(2Πr^{2})

= 9Πr^{2}/16 Πr^{2}

= 0.5625

= 0.5625 (100)

= 56.25%

Question 2 :

The internal and external diameters of a hollow hemispherical vessel are 20 cm and 28 cm respectively. Find the cost to paint the vessel all over at ₹0.14 per cm^{2}

Solution :

internal radius (r) = 20/2 = 10 cm

External radius (R) = 28/2 = 14 cm

Curved surface area of hemisphere = 2Π(R^{2} + r^{2})

= 2 (22/7)(14^{2} + 10^{2})

= 2 (22/7)(196 + 100)

= 1860.57

Cost of painting = ₹0.14 per cm^{2}

Required cost = 0.14 (1860.57)

= ₹260.48

Curved Surface Area and Total Surface Area of Frustum Cone

C.S.A. of a frustum

= Π(R +r)l sq. units

T.S.A. of a frustum

= Π(R + r)l + Π(R^{2}+ r^{2})

where, l = √[h^{2} +(R −r)^{2}]

Question 3 :

Th e frustum shaped outer portion of the table lamp has to be painted including the top part. Find the total cost of painting the lamp if the cost of painting 1 sq.cm is ₹2.

Solution :

Curved surface area of lamp = Π(R +r)l sq. units

R = 12 m, r = 6 m and h = 8 m

l = √[h^{2} +(R −r)^{2}]

l = √[8^{2} +(12 −6)^{2}]

l = √[64 + 6^{2}]

l = √(64 + 36) = √100

l = 10

C.S.A of frustum shaped lamp = Π(R +r)l

Area to be painted = Π(R +r)l + Πr^{2}

= (22/7)[(12 + 6)(10) + 6^{2}]

= (22/7)[180+36]

= 678.85

Cost per sq.cm is ₹2

Required cost = 678.85 (2)

= ₹1357.71

After having gone through the stuff given above, we hope that the students would have understood, "Curved Surface Area and Total Surface Area of Sphere and Hemisphere".

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