Question

Find the circumference of a circle whose area is equal to the sum of the areas of two circles with the radii 7cm and 12cm, give your answer correct to two decimal places.
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Answer 1

Given :

• Area of circle is equal to sum of areas of two circles with the radius of 7 cm and 12 cm.

To find :

• Circumference of circle

Solution :

Let the radius of the two circles be x cm and y cm.

We know that,

Area of circle = πr²

ATQ,

→ Area of circle = π(7)² + π(12)²

→ Area of circle = π(49 + 144)

→ Area of circle = π(193)

→ Area of circle = 193π cm²

Now,

→ πr² = 193π

→ r² = 193π/π

→ r² = 193

→ r = √193

→ r = 13.89 cm (Approx.)

Now we know that,

Circumference of circle = 2πr

→ Circumference of circle = 2 × 3.14 × 13.89

→ Circumference of circle = 87.23 cm (Approx.)

Therefore,

Circumference of circle = 87.23 cm (Approx.)

Answer 2

Answer :-

$\: \: \underline{\boxed{\rm{\blue{\mapsto \: \: Firstly, \: let's \: analyse \: the \: question}}}}$

Here we are given that the Area of the Biggest Circle (which is unknown) is equal to the sum of of the areas of other two circles of radii 7 cm and 12 cm. And then we have to find the Circumference of the biggest circle. Question seems to be simple, but it can become simpler when we start cancelling out the terms from formula itself. Let's do it !

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★ Formulas to be used :-

$\: \: \underline{\boxed{\rm{\green{\Longrightarrow \: \: Area \: of \: circle \: = \: \pi \: \times \: (radius)^{2} \: \: \:[(\pi r^{2})]}}}}$

$\: \: \underline{\boxed{\rm{\green{\Longrightarrow \: \: Circumference \: of \: the \: circle \: = \: 2\pi (radius) \: \: \: [2\pi r]}}}}$

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★ Question :-

Find the circumference of a circle whose area is equal to the sum of the areas of two circles with the radii 7cm and 12cm, give your answer correct to two decimal places.

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★ Solution :-

• Let the circle (⭕) whose Circumference we need to find, be 'A'.

• Let the circle (⭕) whose radius is 7 cm be 'B'.

• Let the circle (⭕) whose radius is 12 cm be 'C'.

Then, its given that,

» Radius of circle B = 7 cm

» Radius of circle C = 12 cm

» Area of circle A = (Area of Circle B) + (Area of Circle C) .

Then let's apply the values, and find the answer.

$\: \: \bf{\orange{\Longrightarrow \: \: \: \pi r^{2}_{(for \: larger \: circle)} \: = \: \pi r^{2}_{(for \: r \: = \: 7)} \: + \: \pi r^{2}_{(for \: r \: = \: 12)}}}$

⌬ πr² (for A) = πr² (for B) + πr² (for C)

⌬ πr² (for A) = π(7)² + π(12)²

⌬ π × r² (for A) = π((7)²+ (12)²)

⌬ π × r² (for A) = π × (49 + 144)

⌬ π × r² (for A) = π × 193

Sending π , another side, we get

$\: \: \: \huge{\red{\bf{\Longrightarrow \: \: \: r_{(for \: circle \: A)} \: = \: \purple{\dfrac{\not{\pi} \: \times 193}{\not{\pi}}}}}}$

⌬ r² = 193

$\: \: \huge{\blue{\bf{\longrightarrow \: \: r \: = \sqrt{193} \: = \: 13.89\: cm \: (approx.)}}}$

• Hence radius of the biggest required circle is = r = 13.89 cm

Now let us find the circumference of this circle.

⌬ Circumference of circle A = 2πr

⌬ Circumference of Circle A = 2 × 3.14 × 13.89

⌬ Circumference of Circle A = 87.229 cm (approx.)

$\: \: \underline{\boxed{\rm{\red{\mapsto \: \: \: Hence, \: the \: required \: circumference \: of \: the \: circle \: is \: \underline{\green{87.229 \: cm}}}}}}$

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$\: \: \underline{\boxed{\sf{\purple{\mapsto \: \: \: Confused? \: Don't \: worry \: let's \: verify \: it}}}}$

For verification, ae need to simply apply the value we got into the equation we formed. Then,

=> πr² = π(7)² + π(12)²

Cancelling π from all the terms, we get,

=> (13.89)² = (7)² + (12)²

=> 193 = 49 + 143

=> 193 = 193

Clearly, LHS = RHS.

Here the condition satisfies, so our answer is correct.

Hence, Verified.

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$\: \: \underline{\boxed{\rm{\pink{Reference \: as \: Supplementary \: Counsel \: :-}}}}$

• Area of Parallelogram = Base × Height

• Area or Square = (Side)²

• Area of Rectangle = Length × Breadth

• Area of Triangle = ½ × Base × Height

• Perimeter of Square = 4 × Side

• Perimeter of Rectangle = 2 × (Length + Breadth)

• Volume of Cuboid = Length × Breadth × Height

• Volume of Cube = (Side)³

• Volume of Cylinder = πr²h

where r is the radius of base and h is the height.

• Volume of Cone = ⅓ × (πr²h)