Find the Surface Area of Cones

The surface area of a cone is the sum of the lateral surface area and the base surface area. If you know the radius of the base and the slant height of the cone, you can easily find the total surface area using a standard formula. Sometimes, however, you might have the radius and some other measurement, such as the height or volume of the cone. In these instances, you can use the Pythagorean Theorem and the volume formula to derive the slant height, and thus the surface area of the cone.

Steps

If You Know the Radius and Slant Height

  1. Set up the formula for the surface area of the cone. The formula is <math>\text{SA} = (\pi)(r)(s) + (\pi)(r^{2})</math>, where <math>\text{SA}</math> equals the surface area of the cone, <math>r</math> equals the length of the radius of the cone’s base, and <math>s</math> equals the slant height of the cone.[1]
    • The total surface area of a cone is equal to the sum of the lateral surface area (<math>(\pi)(r)(s) </math>) and the base area (<math>(\pi)(r^{2})</math>), since the base of a cone is a circle.
    • The slant height is the diagonal distance from the top vertex of the cone to the edge of the base.[2]
    • Make sure you don’t confuse the “slant height” with the “height,” which is the perpendicular distance between the top vertex to the base.[3]
  2. Plug the value of the radius into the formula. This length should be given, or you should be able to measure it. Make sure you substitute for both <math>r</math> variables in the formula.
    • For example, if the radius of the base of a cone is 5 cm, your formula will look like this: <math>\text{SA} = (\pi)(5)(s) + (\pi)(5^{2})</math>.
  3. Plug the value of the slant height into the formula. This length should be given, or you should be able to measure it.
    • For example, if the slant height of a cone is 10 cm, your formula will look like this: <math>\text{SA} = (\pi)(5)(10) + (\pi)(5^{2})</math>.
  4. Calculate the lateral surface area of the cone (<math>(\pi)(r)(s) </math>). To do this, multiply the radius, slant height, and <math>\pi</math>. If you are not using a calculator, use 3.14 as the value of <math>\pi</math>.
    • For example:
      <math>\text{SA} = (\pi)(5)(10) + (\pi)(5^{2})</math>
      <math>\text{SA} = (3.14)(5)(10) + (\pi)(5^{2})</math>
      <math>\text{SA} = 157 + (\pi)(5^{2})</math>
  5. Calculate the area of the cone’s base (<math>(\pi)(r^{2})</math>). To do this, square the radius of the base, then multiply by <math>\pi</math>. If you are not using a calculator, use 3.14 as the value of <math>\pi</math>.
    • For example:
      <math>\text{SA} = 157 + (\pi)(5^{2})</math>
      <math>\text{SA} = 157 + (3.14)(25)</math>
      <math>\text{SA} = 157 + 78.5</math>
  6. Add the lateral surface area and the base area of the cone. This will give you the total surface area of the cone, in square units.
    • For example:
      <math>\text{SA} = 157 + 78.5 = 235.5</math>
      So, the surface area of a cone with a radius of 5 cm and a slant height of 10 cm is 235.5 square centimeters.

If You Know the Radius and the Perpendicular Height

  1. Set up the formula for the Pythagorean Theorem. The formula is <math>a^{2} + b^{2} = c^{2}</math>, where <math>a</math> and <math>b</math> equal the side lengths of a right triangle, and <math>c</math> equals the length of the hypotenuse (the side opposite the right angle).[4]
    • Make sure you don’t confuse the height of the cone with the slant height, which is the diagonal distance from the top vertex of the cone to the edge of the base.[2]
    • The height is the perpendicular distance between the top vertex to the base.[3]
  2. Plug the length of the radius and height into the formula. You will use the radius and height of the cone as the two sides of a right triangle. Substitute the radius for the variable <math>a</math> and the height for the variable <math>b</math>.
    • For example, if the radius of a cone is 5 cm and the height is 12 cm, your formula will look like this: <math>5^{2} + 12^{2} = c^{2}</math>.
  3. Square the lengths of the radius and height, then add. Remember that squaring a number means to multiply it by itself.
    • For example:
      <math>5^{2} + 12^{2} = c^{2}</math>
      <math>25 + 144 = c^{2}</math>
      <math>169 = c^{2}</math>
  4. Take the square root of each side of the equation. This will give you the length of the hypotenuse of the right triangle, which is equal to the the slant height of the cone.[5]
    • For example:
      <math>169 = c^{2}</math>
      <math>\sqrt{169} = \sqrt{c^{2}}</math>
      <math>13 = c</math>
      So, the slant height of the cone is 13 cm.
  5. Set up the formula for the surface area of the cone. The formula is <math>\text{SA} = (\pi)(r)(s) + (\pi)(r^{2})</math>, where <math>\text{SA}</math> equals the surface area of the cone, <math>r</math> equals the length of the radius of the cone’s base, and <math>s</math> equals the slant height of the cone.[1]
    • The total surface area of a cone is equal to the sum of the lateral surface area (<math>(\pi)(r)(s) </math>) and the base area (<math>(\pi)(r^{2})</math>, since the base of a cone is a circle).
  6. Plug all the known values into the formula. The radius should be given, and you already calculated the slant height. Make sure you use the slant height in the surface area formula, not the (perpendicular) height. If you are not using a calculator, use 3.14 for <math>\pi</math>
    • For example, for a cone with a radius of 5 cm and a slant height of 13 cm, your formula will look like this: <math>\text{SA} = (3.14)(5)(13) + (3.14)(5^{2})</math>.
  7. Multiply to find the lateral area and the base area. Then, add these products together. The sum will give you the total surface area of the cone in square units.
    • For example:
      <math>\text{SA} = (3.14)(5)(13) + (3.14)(5^{2})</math>
      <math>\text{SA} = 204.1 + (3.14)(25)</math>
      <math>\text{SA} = 204.1 + 78.5</math>
      <math>\text{SA} = 282.6</math>
      So, the surface area of a cone with a radius of 5 cm and a height of 12 cm is 282.6 square centimeters.

If You Know the Radius and the Volume

  1. Set up the formula for the volume of a cone. The formula is <math>V = \frac{1}{3}(\pi)(r^{2})(h)</math>, where <math>V</math> equals the volume of the cone, <math>r</math> equals the radius of the cone’s base, and <math>h</math> equals the perpendicular height of the cone.[6]
    • Make sure you don’t confuse the height of the cone with the slant height, which is the diagonal distance from the top vertex of the cone to the edge of the base.[2]
    • The height is the perpendicular distance between the top vertex to the base.[3]
  2. Plug the known values into the formula. You should know the volume and the length of the radius. If not, you cannot use this method. If you are not using a calculator, use 3.14 for <math>\pi</math>.
    • For example, if you know a cone has a volume of 950 cubic centimeters and a radius of 6 centimeters, your formula will look like this: <math>950 = \frac{1}{3}(3.14)(6^{2})(h)</math>.
  3. Complete the multiplication. First, square the radius, then multiply that value by <math>\pi</math>. Then, multiply that product by <math>\frac{1}{3}</math>. This will give you the coefficient for the <math>h</math> variable.
    • For example:
      <math>950 = \frac{1}{3}(3.14)(6^{2})(h)</math>
      <math>950 = \frac{1}{3}(3.14)(36)(h)</math>
      <math>950 = \frac{1}{3}(113.04)(h)</math>
      <math>950 = 37.68h</math>
  4. Divide each side by the <math>h</math> coefficient. This will give you the value of <math>h</math>, which is the perpendicular height of the cone. You will need this information to find the slant height of the cone, which is necessary to know when solving for the surface area.
    • For example:
      <math>950 = 37.68h</math>
      <math>\frac{950}{37.68} = \frac{37.68h}{37.68}</math>
      <math>25.21 = h</math>
      So, the height of the cone is 25.21 cm.
  5. Set up the formula for the Pythagorean Theorem. The formula is <math>a^{2} + b^{2} = c^{2}</math>, where <math>a</math> and <math>b</math> equal the side lengths of a right triangle, and <math>c</math> equals the length of the hypotenuse (the side opposite the right angle).[4]
  6. Plug the length of the radius and height into the formula. You will use the radius and height of the cone as the two sides of a right triangle. Substitute the radius for the variable <math>a</math> and the height for the variable <math>b</math>
    • For example, if the radius of a cone is 6 cm and the height is 25.21 cm, your formula will look like this: <math>6^{2} + 25.21^{2} = c^{2}</math>.
  7. Solve for <math>c</math>. This will give you the length of the right triangle’s hypotenuse, which is also the slant height of the cone.
    • For example:
      <math>6^{2} + 25.21^{2} = c^{2}</math>
      <math>36 + 635.54 = c^{2}</math>
      <math>671.54 = c^{2}</math>
      <math>\sqrt{671.54} = \sqrt{c^{2}}</math>
      <math>25.91 = c</math>
      So, the slant height of the cone is 25.91 cm.
  8. Set up the formula for the surface area of the cone. The formula is <math>\text{SA} = (\pi)(r)(s) + (\pi)(r^{2})</math>, where <math>\text{SA}</math> equals the surface area of the cone, <math>r</math> equals the length of the radius of the cone’s base, and <math>s</math> equals the slant height of the cone.[1]
    • The total surface area of a cone is equal to the sum of the lateral surface area (<math>(\pi)(r)(s) </math>) and the base area (<math>(\pi)(r^{2})</math>, since the base of a cone is a circle).
  9. Plug all the known values into the formula. Make sure you use the slant height in the surface area formula, not the (perpendicular) height. If you are not using a calculator, use 3.14 for <math>\pi</math>
    • For example, for a cone with a radius of 6 cm and a slant height of 25.91 cm, your formula will look like this: <math>\text{SA} = (3.14)(6)(25.91) + (3.14)(6^{2})</math>.
  10. Multiply to find the lateral area and the base area. Then, add these products together. The sum will give you the total surface area of the cone in square units.
    • For example:
      <math>\text{SA} = (3.14)(6)(25.91) + (3.14)(6^{2})</math>
      <math>\text{SA} = 488.14 + (3.14)(36)</math>
      <math>\text{SA} = 488.14 + 113.04</math>
      <math>\text{SA} = 601.18</math>
      So, the surface area of a cone with a radius of 6 centimeters and a volume of 950 cubic centimeters is 601.18 square centimeters.



Tips

  • The Pythagorean theorem applies to the radius, perpendicular height, and slant height, with the slant height acting as the hypotenuse: (radius)2 + (perpendicular height)2 = (slant height)2.

Related Articles

Sources and Citations

  1. ↑ 1.0 1.1 1.2 http://www.mathopenref.com/conearea.html
  2. ↑ 2.0 2.1 2.2 http://www.mathwords.com/s/slant_height.htm
  3. ↑ 3.0 3.1 3.2 http://www.mathopenref.com/cone.html
  4. ↑ 4.0 4.1 http://mathworld.wolfram.com/PythagoreanTheorem.html
  5. ↑ http://www.algebralab.org/Word/Word.aspx?file=Geometry_VolumeSurfaceAreaCones.xml
  6. ↑ http://www.mathopenref.com/conevolume.html
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