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# what is the ratio of the total surface area of the solid hemisphere to the square of its radius

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## What is the ratio of the total surface area of the solid hemisphere to the square of its radius.

What is the ratio of the total surface area of the solid hemisphere to the square of its radius.

## What is the ratio of the total surface area of the solid hemisphere to the square of its radius.

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## What is the ratio of the T.S.A of the solid hemi – sphere to the square of its radius.

What is the ratio of the T.S.A of the solid hemi – sphere to the square of its radius.. Ans: Hint: Assume the required ratio of the T.S.A of the solid – hemisphere and the square of its radius as R. Assume that the radius of the solid hemisphere is r...

## What is the ratio of the T.S.A of the solid hemi – sphere to the square of its radius.

Last updated date: 20th Jan 2023

• Total views: 185.4k • Views today: 5.81k Answer Verified 185.4k+ views

Hint: Assume the required ratio of the T.S.A of the solid – hemisphere and the square of its radius as R. Assume that the radius of the solid hemisphere is r. Now, apply the formula for the Total Surface Area of the solid hemisphere given as T.S.A =

3π r 2 3πr2

. Take the expression of T.S.A in the numerator and the square of the radius

( r 2 ) (r2)

in the denominator. Cancel the common factors to get the answer.

Complete step by step solution:

Here we have been provided with a solid hemisphere and we are asked to find the ratio of the Total Surface Area of the hemisphere to the square of its radius. Let us assume the required ratio is R.

Now, let us assume the radius of the hemisphere is r. We know that the Total Surface Area of a solid hemisphere is given by the formula T.S.A =

3π r 2 3πr2

. The square of the radius of the hemisphere will be equal to

( r 2 ) (r2)

. According to the question we have to take the ratio of T.S.A to the square of the radius that means T.S.A will be in the numerator of the fraction while

( r 2 ) (r2)

will be in the denominator. So, mathematically we have the ratio (R) as: -

⇒R= 3π r 2 r 2 ⇒R=3πr2r2

Cancelling the common factors and converting the fraction into the ratio we get,

⇒R= 3π 1 ∴R=3π:1 ⇒R=3π1∴R=3π:1

Hence, the required ratio is

3π:1 3π:1 .

Note: Note that we get a hemisphere when we cut a sphere into two halves. The total surface area of a sphere is

4π r 2 4πr2

so do not think that cutting the sphere into two halves makes the total surface area of the hemisphere half of that of sphere but only the volume gets halved. Remember the formulas for a solid sphere: - Curved Surface Area =

2π r 2 2πr2 and Volume = 2 3 π r 3 23πr3 .

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