Spherical cap

What is the surface area of a spherical cap, the base diameter 27 m, and height 2 m?

Correct answer:

S =  585.12 m2

Step-by-step explanation:

D=27 m h=2 m  ρ=D/2=27/2=227​=13.5 m  r2 = (r−h)2+ρ2 2rh = h2+ρ2  r=2⋅ hh2+ρ2​=2⋅ 222+13.52​=16745​=46.5625 m  S=2π⋅ r⋅ h=2⋅ 3.1416⋅ 46.5625⋅ 2=585.12 m2

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Showing 1 comment:
Dr. Math
To find the surface area of a spherical cap, we'll use the following formula:

A = 2 π r h


where:
- A = surface area of the spherical cap,
- r = radius of the sphere,
- h = height of the cap.

However, we first need to find the radius of the sphere ( r ) using the given base diameter ( D = 27 m ) and height ( h = 2 m ).

Step 1:

Find the Radius of the Sphere ( r )
The relationship between the spherical cap's height ( h ), base radius ( a ), and sphere radius ( r ) is given by:

a2 + (r - h)2 = r2


Given the base diameter D = 27 m , the base radius a = D2 = 13.5 m . Plugging in the values:

(13.5)2 + (r - 2)2 = r2

182.25 + r2 - 4r + 4 = r2

182.25 + 4 - 4r = 0

186.25 = 4r

r = 186.254 = 46.5625 m

Step 2:

Calculate the Surface Area of the Spherical Cap
Now, plug r = 46.5625 m and h = 2 m into the surface area formula:

A = 2 π r h = 2 π (46.5625)(2)

A = 2 π (93.125) ≈ 186.25 π m2


A ≈ 186.25 × 3.1416 ≈ 585.05 m2

Final Answer


186.25π m2 ≈ 585.05 m2







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