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An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.?
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To find the perimeter and area of an equilateral triangle inscribed in a circle with a radius of 20 cm, we can follow these steps:
1. Finding the Side Length of the Triangle
- The radius (R) of the circumcircle of an equilateral triangle is related to the side length (a) by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
- Rearranging the formula gives:
\[ a = R \cdot \sqrt{3} \]
- Substituting the value of R:
\[ a = 20 \cdot \sqrt{3} \]
- Therefore, the side length of the triangle is approximately:
\[ a \approx 20 \cdot 1.732 = 34.64 \text{ cm} \]
2. Calculating the Perimeter
- The perimeter (P) of an equilateral triangle is given by:
\[ P = 3a \]
- Substituting the value of a:
\[ P = 3 \cdot (20 \cdot \sqrt{3}) \]
- Thus,
\[ P \approx 3 \cdot 34.64 \approx 103.92 \text{ cm} \]
3. Calculating the Area
- The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- Substituting the value of a:
\[ A = \frac{\sqrt{3}}{4} (20 \cdot \sqrt{3})^2 \]
- This simplifies to:
\[ A = \frac{\sqrt{3}}{4} \cdot 1200 = 300\sqrt{3} \]
- Therefore, the area is approximately:
\[ A \approx 300 \cdot 1.732 \approx 519.62 \text{ cm}^2 \]
Final Results
- Perimeter: Approximately 103.92 cm
- Area: Approximately 519.62 cm²
1. Finding the Side Length of the Triangle
- The radius (R) of the circumcircle of an equilateral triangle is related to the side length (a) by the formula:
\[ R = \frac{a}{\sqrt{3}} \]
- Rearranging the formula gives:
\[ a = R \cdot \sqrt{3} \]
- Substituting the value of R:
\[ a = 20 \cdot \sqrt{3} \]
- Therefore, the side length of the triangle is approximately:
\[ a \approx 20 \cdot 1.732 = 34.64 \text{ cm} \]
2. Calculating the Perimeter
- The perimeter (P) of an equilateral triangle is given by:
\[ P = 3a \]
- Substituting the value of a:
\[ P = 3 \cdot (20 \cdot \sqrt{3}) \]
- Thus,
\[ P \approx 3 \cdot 34.64 \approx 103.92 \text{ cm} \]
3. Calculating the Area
- The area (A) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
- Substituting the value of a:
\[ A = \frac{\sqrt{3}}{4} (20 \cdot \sqrt{3})^2 \]
- This simplifies to:
\[ A = \frac{\sqrt{3}}{4} \cdot 1200 = 300\sqrt{3} \]
- Therefore, the area is approximately:
\[ A \approx 300 \cdot 1.732 \approx 519.62 \text{ cm}^2 \]
Final Results
- Perimeter: Approximately 103.92 cm
- Area: Approximately 519.62 cm²
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An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.?
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An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.?.
An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for An equilateral triangle is inscribed in a circle touching the three points on the circumference of the circle . If the radius of the circle is 20cm , then find the perimeter and are of the triangle.?.
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