{"id":2800,"date":"2023-05-15T18:05:35","date_gmt":"2023-05-15T18:05:35","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=2800"},"modified":"2024-10-18T20:51:39","modified_gmt":"2024-10-18T20:51:39","slug":"volume-and-surface-area-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/volume-and-surface-area-learn-it-2\/","title":{"raw":"Volume and Surface Area: Learn It 2","rendered":"Volume and Surface Area: Learn It 2"},"content":{"raw":"

Finding the Volume and Surface Area of a Sphere<\/h2>\r\n

A sphere<\/strong> is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below. Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course.<\/p>\r\n

\r\n
\r\n

volume and surface area of a sphere<\/h3>\r\n

For a sphere with radius [latex]r\\text{:}[\/latex]<\/p>\r\n

 <\/p>\r\n

\"A<\/center>\r\n

 <\/p>\r\n

Note: We will approximate [latex]\\pi [\/latex] with [latex]3.14[\/latex].<\/em><\/p>\r\n<\/div>\r\n<\/section>\r\n

It is important to note that for both the volume and surface area of a sphere you use the radius of the sphere. Sometimes questions will only give you the diameter ([latex]d[\/latex]). To find the radius when given the diameter [latex]r = \\frac{d}{2}[\/latex] .<\/section>\r\n
A sphere has a radius [latex]6[\/latex] inches. Find its:\r\n\r\n
    \r\n\t
  1. volume<\/li>\r\n\t
  2. surface area<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"] Step 1 is the same for both 1. and 2., so we will show it just once.\r\n\r\n\r\n\r\n\r\n
    Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n\"A<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n\t
    1. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nthe volume of the sphere<\/td>\r\n<\/tr>\r\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\nlet [latex]V[\/latex] = volume<\/td>\r\n<\/tr>\r\n
      Step 4. Translate.<\/strong> Write the appropriate formula.<\/td>\r\n[latex]V=\\Large\\frac{4}{3}\\normalsize\\pi {r}^{3}[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 5. Solve.<\/strong><\/td>\r\n[latex]V\\approx \\Large\\frac{4}{3}\\normalsize\\left(3.14\\right){6}^{3}[\/latex] [latex]V\\approx 904.32\\text{ cubic inches}[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 6. Check:<\/strong> Double-check your math on a calculator.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
      Step 7. Answer<\/strong> the question.<\/td>\r\nThe volume is approximately [latex]904.32[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n\t
    2. \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nthe surface area of the cube<\/td>\r\n<\/tr>\r\n
      Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\nlet S<\/em> = surface area<\/td>\r\n<\/tr>\r\n
      Step 4. Translate.<\/strong> Write the appropriate formula.<\/td>\r\n[latex]S=4\\pi {r}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 5. Solve.<\/strong><\/td>\r\n[latex]S\\approx 4\\left(3.14\\right){6}^{2}[\/latex] [latex]S\\approx 452.16[\/latex]<\/td>\r\n<\/tr>\r\n
      Step 6. Check:<\/strong> Double-check your math on a calculator<\/td>\r\n <\/td>\r\n<\/tr>\r\n
      Step 7. Answer<\/strong> the question.<\/td>\r\nThe surface area is approximately [latex]452.16[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>\r\n
      [ohm2_question hide_question_numbers=1]6983[\/ohm2_question]<\/section>","rendered":"

      Finding the Volume and Surface Area of a Sphere<\/h2>\n

      A sphere<\/strong> is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below. Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course.<\/p>\n

      \n
      \n

      volume and surface area of a sphere<\/h3>\n

      For a sphere with radius [latex]r\\text{:}[\/latex]<\/p>\n

       <\/p>\n

      \"A<\/div>\n

       <\/p>\n

      Note: We will approximate [latex]\\pi[\/latex] with [latex]3.14[\/latex].<\/em><\/p>\n<\/div>\n<\/section>\n

      It is important to note that for both the volume and surface area of a sphere you use the radius of the sphere. Sometimes questions will only give you the diameter ([latex]d[\/latex]). To find the radius when given the diameter [latex]r = \\frac{d}{2}[\/latex] .<\/section>\n
      A sphere has a radius [latex]6[\/latex] inches. Find its:<\/p>\n
        \n
      1. volume<\/li>\n
      2. surface area<\/li>\n<\/ol>\n