Solid Figures

Read till taught in 8th

Cone

Solid Cone

cone



Let h be the height, l be the slant height and r be the radius of a right circular cone then

Curved (lateral) surface area of the solid cone

Curved surface area = πr√(r2 + h2)

Total surface area of a solid cone

Total surface area = curved surface area + area of the circular base
= πrl + πr2
= πr(l+r)

Volume of a solid cone

Volume
= (1/3)πr2h


Cylinder

Solid Cylinder

cylinder



Let h be the height and r be the radius of a solid cylinder then

Curved surface area of a solid cylinder

Curved surface area = perimeter of cross-section × height
= 2πrh

Total surface area of a solid cylinder

Total surface area = curved surface area + area of two circular ends
= 2πrh + 2πr2
= 2πr(h+r)

Volume of a solid cylinder

Volume = area of cross-section × height
= πr2×h

Hollow Cylinder


hollow cylinder


Let r and R be the internal and external radii of the hollow cylinder then
Thickness of the cylinder = R − r
Area of the cross-section = π(R2 − r2)

Curved surface area of the hollow cylinder

Internal Curved surface area = perimeter of internal cross-section × height
= 2πrh
External Curved surface area = perimeter of external cross-section × height
= 2πRh

Total surface area of the hollow cylinder

Total surface area = Internal curved surface area + External curved surface area + area of two ends
= 2πrh + 2πRh + 2π(R2 − r2)
= 2π(rh + Rh + R2 − r2)

Volume of the hollow cylinder

Volume = area of cross-section × height
= π(R2 − r2)×h


Sphere

Solid Sphere

solid sphere



Let r be the radius of a solid sphere then

Surface area of a solid sphere

Surface area = 4πr2

Volume of the solid sphere

Volume = (4/3)πr3

Hemisphere


hemisphere


Let r be the radius of the solid hemisphere then

Curved Surface area of a solid hemisphere

Curved Surface area = 2πr2

Total Surface area of the solid hemisphere

Total Surface area = curved surface area + area of circular base
= 2πr2 + πr2 = 3πr2

Volume of the solid hemisphere

Volume = (1/2)(4/3)πr3 = (2/3)πr3

Spherical Shell


spherical shell


Let r and R be the inner and outer radii of the spherical shell then
Thickness of the shell = R − r

Volume of the spherical shell

Volume = (4/3)π(R3 − r3)

Hemispherical Shell


hemispherical shell


Let r and R be the inner and outer radii of the spherical shell then
Area of base = π(R2 − r2)

Surface area of the hemispherical shell

External Curved Surface area = 2πR2
Internal Curved Surface area = 2πr2

Total Surface area of the hemispherical shell

Total Surface area = 2πR2+ 2πr2 + π(R2 − r2)
= π(3R2 + r2)

Volume of the hemispherical shell

Volume of the hemispherical shell = (2/3)π(R3 − r3)
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