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How Do You Find the Lateral and Surface Areas of a Cone if You Don't Know the Slant Height?

Find the lateral and surface areas of the given cone

Summary

  1. 'r' is the radius of the base, which is 3 inches
  2. 'h' is the height of the cone, which is 4 inches
  3. 'l', the slant height, is 5 inches
  4. π, or pi, is an irrational number that can be approximated as 3.14
  5. 'B' is the area of the base, which is a circle
  6. 'S', the surface area, is the lateral area 'L' plus the area of the base 'B'

Notes

    1. The lateral area of a cone is πrl, where 'r' is the radius of the base and 'l' is the slant height
    2. We are given the radius of the base, but not the slant height
    3. The slant height is the distance from the tip of the cone to a point on the edge of the base
    4. The height we are given is the height of the cone, not the slant height
    1. We need to find the slant height in order to find the lateral area of the cone
    1. The radius of the base, 3 in, and the height of the cone, 4 in, are the legs of the right triangle
    2. The slant height, which we are trying to find, is the hypotenuse
    1. We can use the Pythagorean Theorem to find the missing hypotenuse, 'l'
    2. The radius, 'r', and the height, 'h', are the legs of the right triangle we made, so we can replace 'a' and 'b' with those variables
    3. Then we can plug in 3 for 'r' and 4 for 'h' to solve for 'l'
    4. 32 is 9, and 42 is 16
    5. 9+16 is 25, and the square root of 25 is 5, so we find that 'l' is 5 in
    1. The lateral area is the area of everything that is not a base
    1. π, or pi, is an irrational number that can be approximated as 3.14
    2. Plug 3 in for 'r' and 5 in for 'l' into the formula, L=πrl
    3. Using 3.14 for π, we get L=(3.14)(3)(5)
    4. 3•5=15, and 15•3.14=47.1
    5. So L, the lateral area of the cone, is 47.1 inches squared
    1. The surface area is the lateral area plus the area of the base
    2. We just found the lateral area, but we need to find the area of the base before we can find the surface area
    1. The radius of the circular base is the same as the radius of the cone, 3 inches
    2. So plugging in 3.14 for π and 3 in for 'r' we get B=(3.14)(3)2
    3. 32 is 9, and 9•3.14 is 28.26
    4. So the area of the base is 28.26 inches squared
    1. 'S', the surface area, is the lateral area 'L' plus the area of the base 'B'
    1. 'L', the lateral area, was 47.1 in2
    2. 'B', the area of the base, was 28.26 in2
    3. So 'S', the surface area, is 47.1+28.26, or 75.36 in2