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Math Help - Interpretation of Simplified Formula for Area of a Sector (1/2)(theta)r^2

  1. #1
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    Interpretation of Simplified Formula for Area of a Sector (1/2)(theta)r^2

    Hello everyone,

    I fully understand the idea behind the formula for the area of a sector of a circle with angle  \theta :


     \text{Area of a sector of a circle}=\frac{\theta }{2\pi }\left( \pi {{r}^{2}} \right) \text{         (*)        } ,

     \text{where }\frac{\theta }{2\pi }\text{ gives the specific ''slice'' of the circle}

     \text{and }\pi {{r}^{2}}=\text{area of the circle}\text{.}


    However, I am having trouble interpreting the simplified form of this formula:

     \text{Area of a sector of a circle} = \frac{1}{2} \theta r^2  .

    What does it mean to multiply the angle of the sector by the radius squared, and then dividing this by half? How can this be interpreted to give us the area of a sector of a circle, WITHOUT referring to the original expression  \text{ie (*)}

    Thank you very much.
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  2. #2
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    Re: Interpretation of Simplified Formula for Area of a Sector (1/2)(theta)r^2

    Quote Originally Posted by scherz0 View Post
    Hello everyone,

    I fully understand the idea behind the formula for the area of a sector of a circle with angle  \theta :

     \text{Area of a sector of a circle}=\frac{\theta }{2\pi }\left( \pi {{r}^{2}} \right) \text{         (*)        } ,

     \text{where }\frac{\theta }{2\pi }\text{ gives the specific ''slice'' of the circle}

     \text{and }\pi {{r}^{2}}=\text{area of the circle}\text{.}

    However, I am having trouble interpreting the simplified form of this formula:

     \text{Area of a sector of a circle} = \frac{1}{2} \pi r^2  .

    What does it mean to multiply the angle of the sector by the radius squared, and then dividing this by half? How can this be interpreted to give us the area of a sector of a circle, WITHOUT referring to the original expression  \text{ie (*)}

    Thank you very much.
    first off, you've posted the formula incorrectly. it should be ...

    A = \frac{\theta \cdot r^2}{2}

    to justify this formula w/o using the mentioned geometrical relationship requires the use of the calculus ...

    consider a sector with a very small \theta = d\theta, and note the area of a small sector is close to the area of a triangle.

    remembering the area of a triangle, A = \frac{bh}{2} , the base is the small arclength equal to r \cdot d\theta and the height is equal to r ...

    a small sector of the large sector has area dA = \frac{bh}{2} = \frac{r \cdot d\theta \cdot r}{2} = \frac{r^2 \cdot d\theta}{2}

    suffice it to say that summing all the small sectors that make up the larger sector results in A = \frac{r^2 \cdot \theta}{2}
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  3. #3
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    Re: Interpretation of Simplified Formula for Area of a Sector (1/2)(theta)r^2

    Quote Originally Posted by scherz0 View Post
    Hello everyone,

    I fully understand the idea behind the formula for the area of a sector of a circle with angle  \theta :


     \text{Area of a sector of a circle}=\frac{\theta }{2\pi }\left( \pi {{r}^{2}} \right) \text{         (*)        } ,

     \text{where }\frac{\theta }{2\pi }\text{ gives the specific ''slice'' of the circle}

     \text{and }\pi {{r}^{2}}=\text{area of the circle}\text{.}


    However, I am having trouble interpreting the simplified form of this formula:

     \text{Area of a sector of a circle} = \frac{1}{2} \theta r^2  .

    What does it mean to multiply the angle of the sector by the radius squared, and then dividing this by half? How can this be interpreted to give us the area of a sector of a circle, WITHOUT referring to the original expression  \text{ie (*)}

    Thank you very much.
    Surely you can see that \displaystyle \frac{\theta}{2\pi}\left(\pi r^2\right) = \frac{\pi r^2 \theta}{2\pi} = \frac{r^2 \theta}{2} = \frac{1}{2}r^2 \theta, it's just the cancelling of \displaystyle \pi in the numerator and denominator...
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  4. #4
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    Re: Interpretation of Simplified Formula for Area of a Sector (1/2)(theta)r^2

    Thank you for your responses, skeeter and Prove It.

    @skeeter: Thank you very much for your explanation.

    I hadn't thought to approximate the aforesaid area using a triangle.

    @Prove It: I understand that, but my original question was on how to

    interpret \frac{1}{2}\theta r^2 without at all using  (*) .
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