derivation of all formulas of cone
Answers
Answer:
Step-by-step explanation:
This shaded section is actually part of a larger circle that has a radius of s, the slant height of the cone. (To flatten it, the cone was cut along the red lines, the length of this cut is the slant height of the cone.)
The area of the larger circle is therefore the area of a circle radius s, or
π s 2
The circumference of the larger circle, radius s is
2 π s
The arc AB originally wrapped around the base of the cone, and so its length is the circumference of the base. Recall that circumference of a circle is given by
2 π r
Where r is the radius of the base of the cone.
The ratio of area x of the shaded sector to the area of the whole circle, is the same as the ratio of the arc AB to circumference of the whole circle. To see why this is so, see Area of a circle sector.
Put as an equation
x
area of large circle
=
AB
circumference of large circle
Substituting the values from above:
x
π s 2
=
2 π r
2 π s
Canceling the 2π on the right and solving for x we get
x = π r s
Finally, adding the areas of the base and the top part produces the final formula:
area = π r s + π r 2