Finding the surface area of a cone Finding the surface area of a cone The first step in finding the surface area of a cone is to measure the radius of the circle part of the cone. The next step is to find the area of the circle, or base. The area of a circle is 3.14 times the radius squared (πr2). Now, you will need to find the area of the cone itself. In order to do this, you must measure the side (slant height) of the cone. Make sure you use the same form of measurement as the radius. You can now use the measurement of the side to find the area of the cone. The formula for the area of a cone is 3.14 times the radius times the side (πrl). So the surface area of the cone equals the area of the circle plus the area of the cone and the final formula is given by: SA = πr2 + πrl Where, r is the radius h is the height l is the slant height The area of the curved (lateral) surface of a cone = πrl
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Finding the surface area of cones is not that hard. But it can require some patience and ingenuity, depending on what information is available at the beginning of the problem. Below are some suggested steps to keep track of everything. The surface area of a cone can be derived from the surface area of a square pyramid Start with a square pyramid and just keep increasing the number of sides of the base. After a very large number of sides, you can see that the figure will eventually look like a cone. This observation is important because we can use the formula of the surface area of a square pyramid to find that of a cone l is the slant height. The area of the square is s2 The area of one triangle is (s × l)/2 Since there are 4 triangles, the area is 4 × (s × l)/2 = 2 × s × l Therefore, the surface area, call it SA is: SA = s2 + 2 × s × l : Generally speaking, to find the surface area of any regular pyramid whose base is A, the perimeter is P, and the slant height is l, we use the following formula: Learn More :- Transformations on the coordinate plane
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S = A + 1/2 (P × l) Again A is the area of the base. For a figure with 4 sides, A = s2 with s = length of one side Where does the 1/2 (P × l) come from? Let s be the length of the base of a regular pyramid. Then, the area of one triangle is (s × l)/2 For n triangles and this also means that the base of the pyramid has n sides, we get, ( n × s × l)/2 Now P = n × s. When n = 4, of course, P = 4 × s as already shown. Therefore, after replaning n × s by P, we get S = A + 1/2 (P × l) Let us now use this fact to derive the formula of the surface area of a cone For a cone, the base is a circle, A = π × r2 P=2×π×r To find the slant height, l, just use the Pythagorean Theorem l = r2 + h2 l = √ (r2 + h2) Putting it all together, we get: S = A + 1/2 (P × l) S = π × r2 + 1/2 ( 2 × π × r × √ (r2 + h2) S = π × r2 + π × r × √ (r2 + h2) Tutorcircle.com
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