Question

a hemisphere is cut out from one face of a cubical wodden block such that the diameter of the hemisphere is equal to the length of the cube. determine the surface area of the remaining solid

Answer 1

If a hemisphere is cut out from one face, the surface area of remaining solid = total surface area of cube - base area of hemisphere + lateral surface area of hemisphere

Let the side of cube = a
radius of hemisphere = a/2

TSA of cube = 6a²
base area of hemisphere = π(a/2)² = πa²/4
LSA of hemisphere = 2π(a/2)² = πa²/2

So surface area of remaining solid = 6a² - πa²/4 + πa²/2
= 6a² + πa²/4
= (24a² + πa²) /4

= $\frac{24+ \pi}{4}\ a^2$

Surface area is  $\frac{24+ \pi}{4}\ a^2$

Answer 2

Answer:

a²( 6 + π/4 ) sq. units

Step-by-step explanation:

Let the edge of the cubical wooden block be ' a' units

Total Surface area of the cubical wooden block = 6a² sq. units

Hemisphere is cut out from one face of a cubical wooden block such that diameter of it is equal to edge of the cubical wooden block

Diameter of the hemisphere cut out = ' a ' units

Radius of the hemisphere cut out r = ' a/2 ' units

To determine the total surface area of the remaining solid we need to subtract the area of circular region formed when the hemisphere is cut out and then add the curved surface area of the hemisphere

Area of circular region formed when hemisphere is cut out = πr² = π × ( a/2 )² = π × a²/4 = πa²/4 sq. units

Curved surface area of the hemisphere cut out = 2πr² = 2π × ( a/2 )² = 2πa²/4 sq. units

Total surface area of the remaining solid = Total surface area of the wooden cubical block - Area of circular region formed when hemisphere is cut out + Curved surface area of the hemisphere cut out

= 6a² - πa²/4 + 2πa²/4

= 6a² + πa²/4

= a²( 6 + π/4 ) sq. units

Therefore the total surface area of the remaining solid is a²( 6 + π/4 ) sq. units.