Question

Find the area of the region between the two concentric circles is the length of the chord of the outer circle just touching the inner circle at a point on it is 10 cm

Answer 1

area of the region between the two concentric circles  =  78.5  cm²

Step-by-step explanation:

Let say Radius of inner circle = r

& Outer Circle = R

Let say Say Chord AB touch inner circle at P & Center is O

Now in Δ OAP  & ΔOBP

OA = OB = R  ( Radius of outer circle)

OP = OP (common)

∠OPA = ∠OPB = 90°

=> AP = BP

=> AP + BP = AB = 10 cm

=> AP = BP = 5 cm

Now in Δ OAP

=> OP² = OA² - AP²

OP = r ( radius of inner circle)

=> r² = R² - 5²

=> R² - r² = 25

Area of Outer circle = πR²

Area of inner circle = πr²

the area of the region between the two concentric circles  =  πR² - πr²

=  π (R² - r²)

Using R² - r² = 25

= 25π

using π = 3.14

= 78.5  cm²

area of the region between the two concentric circles  =  78.5  cm²

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Answer 2

The area between two concentric circle is 25π cm²

Explanation:

The area between two concentric circle is difference of area of outer circle and inner circle.

Let the radius of outer circle be R cm and inner circle be r cm

Tangent of inner circle is chord of outer circle.

• For inner circle: -

Radius is perpendicular to tangent.

Therefore, OP is perpendicular to AB

• For outer circle:-

If a line perpendicular to chord from center then line bisect the chord.

Therefore, AP = BP

But AP + BP = 10

2BP = 10

BP = 5 cm

In ΔOPB, ∠OPB = 90°

$OB^2=OP^2+BP^2$

$R^2=r^2=5^2$

$R^2-r^2=25$

Area of outer circle, $A_o=\pi R^2$

Area of inner circle, $A_i=\pi r^2$

Required area $=A_o-A_i$

                        $=\pi R^2-\pi r^2$

                        $=\pi(R^2-r^2)$

substitute $R^2-r^2=25$

Required area $25\pi\text{ cm}^2$

Hence, the area between two concentric circle is 25π cm²

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