{"id":1795,"date":"2025-07-28T19:39:51","date_gmt":"2025-07-28T19:39:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1795"},"modified":"2025-08-13T03:03:15","modified_gmt":"2025-08-13T03:03:15","slug":"arcs-and-sectors-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arcs-and-sectors-learn-it-2\/","title":{"raw":"Arcs and Sectors: Learn It 2","rendered":"Arcs and Sectors: Learn It 2"},"content":{"raw":"

Finding the Area of a Sector of a Circle<\/h2>\r\n
In addition to arc length, we can also use angles to find the area of a sector of a circle<\/strong>. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie.<\/section>
The area of a circle with radius [latex]r[\/latex] can be found using the formula [latex]A=\\pi {r}^{2}[\/latex].<\/section>If the two radii form an angle of [latex]\\theta [\/latex], measured in radians, then [latex]\\frac{\\theta }{2\\pi }[\/latex] is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the <\/span>area of a sector<\/strong> is the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex]\u00a0multiplied by the entire area. (Always remember that this formula only applies if [latex]\\theta [\/latex] is in radians.)<\/span>\r\n\r\n<\/section>
\r\n
[latex]\\begin{align}\\text{Area of sector}&=\\left(\\frac{\\theta }{2\\pi }\\right)\\pi {r}^{2} \\\\ &=\\frac{\\theta \\pi {r}^{2}}{2\\pi } \\\\ &=\\frac{1}{2}\\theta {r}^{2} \\end{align}[\/latex]<\/div>\r\n<\/section>\r\n
\r\n

area of a sector<\/h3>\r\nThe area of a sector<\/strong> of a circle with radius [latex]r[\/latex] subtended by an angle [latex]\\theta [\/latex], measured in radians, is\r\n
\r\n

[latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"442\"]\"Graph The area of the sector equals half the square of the radius times the central angle measured in radians.[\/caption]\r\n\r\n<\/div>\r\n<\/section>

How To: Given a circle of radius [latex]r[\/latex], find the area of a sector defined by a given angle [latex]\\theta [\/latex].<\/strong>\r\n
    \r\n \t
  1. If necessary, convert [latex]\\theta [\/latex] to radians.<\/li>\r\n \t
  2. Multiply half the radian measure of [latex]\\theta [\/latex] by the square of the radius [latex]r:\\text{ } A=\\frac{1}{2}\\theta {r}^{2}[\/latex].<\/li>\r\n<\/ol>\r\n<\/section><\/div>\r\n
    An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees. What is the area of the sector of grass the sprinkler waters?\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Illustration The sprinkler sprays 20 ft within an arc of 30\u00b0.[\/caption]\r\n\r\n[reveal-answer q=\"785904\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785904\"]\r\n\r\nFirst, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:\r\n

    [latex]\\begin{align}30\\text{ degrees}&=30\\cdot \\frac{\\pi }{180} \\\\ &=\\frac{\\pi }{6}\\text{ radians} \\end{align}[\/latex]<\/p>\r\nThe area of the sector is then\r\n

    [latex]\\begin{align}\\text{Area} &= \\frac{1}{2}\\left(\\frac{\\pi }{6}\\right){\\left(20\\right)}^{2} \\\\ &\\approx 104.72 \\end{align}[\/latex]<\/p>\r\nSo the area is about [latex]104.72{\\text{ ft}}^{2}[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

    In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.[reveal-answer q=\"495928\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"495928\"]1.88[\/hidden-answer]<\/section><\/section><\/section>","rendered":"

    Finding the Area of a Sector of a Circle<\/h2>\n
    \n
    In addition to arc length, we can also use angles to find the area of a sector of a circle<\/strong>. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie.<\/section>\n
    \n
    The area of a circle with radius [latex]r[\/latex] can be found using the formula [latex]A=\\pi {r}^{2}[\/latex].<\/section>\n

    If the two radii form an angle of [latex]\\theta[\/latex], measured in radians, then [latex]\\frac{\\theta }{2\\pi }[\/latex] is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the <\/span>area of a sector<\/strong> is the fraction [latex]\\frac{\\theta }{2\\pi }[\/latex]\u00a0multiplied by the entire area. (Always remember that this formula only applies if [latex]\\theta[\/latex] is in radians.)<\/span><\/p>\n<\/section>\n

    \n
    [latex]\\begin{align}\\text{Area of sector}&=\\left(\\frac{\\theta }{2\\pi }\\right)\\pi {r}^{2} \\\\ &=\\frac{\\theta \\pi {r}^{2}}{2\\pi } \\\\ &=\\frac{1}{2}\\theta {r}^{2} \\end{align}[\/latex]<\/div>\n<\/section>\n
    \n
    \n

    area of a sector<\/h3>\n

    The area of a sector<\/strong> of a circle with radius [latex]r[\/latex] subtended by an angle [latex]\\theta[\/latex], measured in radians, is<\/p>\n

    \n

    [latex]A=\\frac{1}{2}\\theta {r}^{2}[\/latex]<\/p>\n

    \"Graph
    The area of the sector equals half the square of the radius times the central angle measured in radians.<\/figcaption><\/figure>\n<\/div>\n<\/section>\n
    How To: Given a circle of radius [latex]r[\/latex], find the area of a sector defined by a given angle [latex]\\theta[\/latex].<\/strong><\/p>\n
      \n
    1. If necessary, convert [latex]\\theta[\/latex] to radians.<\/li>\n
    2. Multiply half the radian measure of [latex]\\theta[\/latex] by the square of the radius [latex]r:\\text{ } A=\\frac{1}{2}\\theta {r}^{2}[\/latex].<\/li>\n<\/ol>\n<\/section>\n<\/div>\n
      \n
      An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees. What is the area of the sector of grass the sprinkler waters?<\/p>\n
      \"Illustration
      The sprinkler sprays 20 ft within an arc of 30\u00b0.<\/figcaption><\/figure>\n