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An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle?
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An equilateral triangle ABC is inscribed in a circle and another circl...
Explanation:
Given: Area of inner circle = 20π
Step 1: Finding the radius of the inner circle
- The area of a circle is given by the formula A = πr^2, where r is the radius.
- Given that the area of the inner circle is 20π, we can find the radius by equating the formula: 20π = πr^2.
- Solving for r, we get r = √20 = 2√5.
Step 2: Finding the side length of the equilateral triangle
- In an equilateral triangle, the radius of the inscribed circle is related to the side length by the formula: r = (s√3) / 6, where s is the side length.
- Substituting the radius we found in Step 1, we get 2√5 = (s√3) / 6.
- Solving for s, we get s = 4√5.
Step 3: Finding the radius of the outer circle
- The radius of the outer circle is equal to the circumradius of the equilateral triangle, which is given by R = s / √3, where s is the side length.
- Substituting the side length we found in Step 2, we get R = 4√5 / √3 = 4√15 / 3.
Step 4: Finding the area of the outer circle
- The area of a circle is given by the formula A = πR^2, where R is the radius.
- Substituting the radius we found in Step 3, we get the area of the outer circle as π(4√15 / 3)^2 = 20π.
Therefore, the area of the outer circle is 20π.
Given: Area of inner circle = 20π
Step 1: Finding the radius of the inner circle
- The area of a circle is given by the formula A = πr^2, where r is the radius.
- Given that the area of the inner circle is 20π, we can find the radius by equating the formula: 20π = πr^2.
- Solving for r, we get r = √20 = 2√5.
Step 2: Finding the side length of the equilateral triangle
- In an equilateral triangle, the radius of the inscribed circle is related to the side length by the formula: r = (s√3) / 6, where s is the side length.
- Substituting the radius we found in Step 1, we get 2√5 = (s√3) / 6.
- Solving for s, we get s = 4√5.
Step 3: Finding the radius of the outer circle
- The radius of the outer circle is equal to the circumradius of the equilateral triangle, which is given by R = s / √3, where s is the side length.
- Substituting the side length we found in Step 2, we get R = 4√5 / √3 = 4√15 / 3.
Step 4: Finding the area of the outer circle
- The area of a circle is given by the formula A = πR^2, where R is the radius.
- Substituting the radius we found in Step 3, we get the area of the outer circle as π(4√15 / 3)^2 = 20π.
Therefore, the area of the outer circle is 20π.
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An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle?
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An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle?.
An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle ABC is inscribed in a circle and another circle is inscribed in the triangle ABC, of the area of the inner circle is 20 π, then what is the area of the outer circle?.
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