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The perimeters of a circle, a square, and an equilateral triangle are equal. Which one of the following statements is true?
- a)The circle has the largest area.
- b)The square has the largest area.
- c)The equilateral triangle has the largest area.
- d)All the three shapes have the same area.
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
The perimeters of a circle, a square, and an equilateral triangle are ...
The perimeter of Circle of radius r = 2πr
The perimeter of a square of side s = 4s
The perimeter of an equilateral triangle of side a = 3a
The perimeters of a circle, a square, and an equilateral triangle are equal.
2πr = 4s = 3a
The perimeter of a square of side s = 4s
The perimeter of an equilateral triangle of side a = 3a
The perimeters of a circle, a square, and an equilateral triangle are equal.
2πr = 4s = 3a
- s = 3/4 a = 0.75 a
- r = 3/2π a = 0.477 a
Area of an equilateral triangle:
Area of square = s2 = (0.75 a)2 = 0.5625 a2
Area of Circle = 2πr2 = π(0.477 a)2 = 0.714 a2
Acircle > ASquare > Atrg
Area of square = s2 = (0.75 a)2 = 0.5625 a2
Area of Circle = 2πr2 = π(0.477 a)2 = 0.714 a2
Acircle > ASquare > Atrg
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The perimeters of a circle, a square, and an equilateral triangle are ...
Explanation:
Perimeter of the Shapes:
- Let's denote the perimeter of all three shapes as P.
- Perimeter of a circle = 2πr, where r is the radius of the circle.
- Perimeter of a square = 4s, where s is the side length of the square.
- Perimeter of an equilateral triangle = 3s, where s is the side length of the equilateral triangle.
Area of the Shapes:
- Area of a circle = πr²
- Area of a square = s²
- Area of an equilateral triangle = √3/4 * s² (where s is the side length of the equilateral triangle)
Comparison:
- Given that the perimeters of all three shapes are equal, we have P = 2πr = 4s = 3s.
- From this equation, we can find that r = 2s/π.
- Now, we can compare the areas of the shapes.
- Area of the circle = π(2s/π)² = 4s²/π
- Area of the square = s²
- Area of the equilateral triangle = √3/4 * s²
Conclusion:
- Comparing the areas of the shapes, we find that the area of the circle (4s²/π) is greater than the area of the square (s²) and the area of the equilateral triangle (√3/4 * s²).
- Therefore, the statement "The circle has the largest area" is true.
Perimeter of the Shapes:
- Let's denote the perimeter of all three shapes as P.
- Perimeter of a circle = 2πr, where r is the radius of the circle.
- Perimeter of a square = 4s, where s is the side length of the square.
- Perimeter of an equilateral triangle = 3s, where s is the side length of the equilateral triangle.
Area of the Shapes:
- Area of a circle = πr²
- Area of a square = s²
- Area of an equilateral triangle = √3/4 * s² (where s is the side length of the equilateral triangle)
Comparison:
- Given that the perimeters of all three shapes are equal, we have P = 2πr = 4s = 3s.
- From this equation, we can find that r = 2s/π.
- Now, we can compare the areas of the shapes.
- Area of the circle = π(2s/π)² = 4s²/π
- Area of the square = s²
- Area of the equilateral triangle = √3/4 * s²
Conclusion:
- Comparing the areas of the shapes, we find that the area of the circle (4s²/π) is greater than the area of the square (s²) and the area of the equilateral triangle (√3/4 * s²).
- Therefore, the statement "The circle has the largest area" is true.
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The perimeters of a circle, a square, and an equilateral triangle are equal. Which one of the following statements is true?a)The circle has the largest area.b)The square has the largest area.c)The equilateral triangle has the largest area.d)All the three shapes have the same area.Correct answer is option 'A'. Can you explain this answer?
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