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Sagot :
Certainly! Let's examine each of the given statements about the area of a circle and determine which are true.
1. Area [tex]\( = \pi d \)[/tex]
- This statement is false. The correct formula for the area of a circle involves the radius [tex]\( r \)[/tex], not the diameter [tex]\( d \)[/tex]. The diameter is related to the radius by the equation [tex]\( d = 2r \)[/tex], but this specific formula is incorrect for calculating the area of a circle.
2. Area [tex]\( = \pi r^2 \)[/tex]
- This statement is true. The correct formula for the area of a circle is given by [tex]\( \text{Area} = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
3. The area formula can be found by breaking apart the circle and forming a triangle.
- This statement is true. One method of deriving the area of a circle is by conceptually cutting the circle into sectors and rearranging these into a shape that approximates a triangle. The area of this triangle, with base equal to the circumference ([tex]\( 2\pi r \)[/tex]) and height equal to the radius ([tex]\( r \)[/tex]), leads to the area formula [tex]\( \pi r^2 \)[/tex].
4. The area formula can be found by breaking apart the circle and forming a parallelogram.
- This statement is true. Another method involves slicing the circle into narrow sectors that can be rearranged into a shape resembling a parallelogram. The area can be calculated similarly and also leads to [tex]\( \pi r^2 \)[/tex].
5. The area of a circle is in square units.
- This statement is true. Since the area measures the amount of two-dimensional space inside the circle, it is expressed in square units, such as square meters ([tex]\( m^2 \)[/tex]), square centimeters ([tex]\( cm^2 \)[/tex]), etc.
So, to summarize, the statements about the area of a circle that are true are:
- Area [tex]\( = \pi r^2 \)[/tex].
- The area formula can be found by breaking apart the circle and forming a triangle.
- The area formula can be found by breaking apart the circle and forming a parallelogram.
- The area of a circle is in square units.
Thus, the responses to the statements given in the problem are:
- Area [tex]\( = \pi d \)[/tex]: False
- Area [tex]\( = \pi r^2 \)[/tex]: True
- The area formula can be found by breaking apart the circle and forming a triangle: True
- The area formula can be found by breaking apart the circle and forming a parallelogram: True
- The area of a circle is in square units: True
1. Area [tex]\( = \pi d \)[/tex]
- This statement is false. The correct formula for the area of a circle involves the radius [tex]\( r \)[/tex], not the diameter [tex]\( d \)[/tex]. The diameter is related to the radius by the equation [tex]\( d = 2r \)[/tex], but this specific formula is incorrect for calculating the area of a circle.
2. Area [tex]\( = \pi r^2 \)[/tex]
- This statement is true. The correct formula for the area of a circle is given by [tex]\( \text{Area} = \pi r^2 \)[/tex], where [tex]\( r \)[/tex] is the radius of the circle.
3. The area formula can be found by breaking apart the circle and forming a triangle.
- This statement is true. One method of deriving the area of a circle is by conceptually cutting the circle into sectors and rearranging these into a shape that approximates a triangle. The area of this triangle, with base equal to the circumference ([tex]\( 2\pi r \)[/tex]) and height equal to the radius ([tex]\( r \)[/tex]), leads to the area formula [tex]\( \pi r^2 \)[/tex].
4. The area formula can be found by breaking apart the circle and forming a parallelogram.
- This statement is true. Another method involves slicing the circle into narrow sectors that can be rearranged into a shape resembling a parallelogram. The area can be calculated similarly and also leads to [tex]\( \pi r^2 \)[/tex].
5. The area of a circle is in square units.
- This statement is true. Since the area measures the amount of two-dimensional space inside the circle, it is expressed in square units, such as square meters ([tex]\( m^2 \)[/tex]), square centimeters ([tex]\( cm^2 \)[/tex]), etc.
So, to summarize, the statements about the area of a circle that are true are:
- Area [tex]\( = \pi r^2 \)[/tex].
- The area formula can be found by breaking apart the circle and forming a triangle.
- The area formula can be found by breaking apart the circle and forming a parallelogram.
- The area of a circle is in square units.
Thus, the responses to the statements given in the problem are:
- Area [tex]\( = \pi d \)[/tex]: False
- Area [tex]\( = \pi r^2 \)[/tex]: True
- The area formula can be found by breaking apart the circle and forming a triangle: True
- The area formula can be found by breaking apart the circle and forming a parallelogram: True
- The area of a circle is in square units: True
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