How to Find the Formula for Surface Area of a Cone
Area of Circle = πs²
Circumference of Circle = 2πs
s = slant height of Cone
Circumference of Cone = 2πr
x = Lateral Area of Cone
r = radius of Cone
h = height of Cone
s
s
h
r
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“s” works as the radius for the circle. So the area is πs² rather than πr²
The perimeter of the lateral area of the cone includes both s and the curve connecting them (the
curve would be the circumference of the cone). Everything inside makes up the lateral area of the
cone
Step 1: Formula (x represents the lateral area of the cone)
x / Area of Circle = Circumference of Cone / Circumference of Circle
x / πs² = 2πr / 2πs
Step 2: Simplify
2π cancels out:
x / πs² = r / s
Multiply πs² to each side: x = πrs
Step 3: Fill in “s” and add base area
If you look at the lateral area of the cone above, you see the slant height of the cone is a
hypotenuse to the height and radius of the cone. So, if you use the Pythagoream Theorum, the
slant height is equal to √ (r² + h²).
Lateral Area = πr√ (r² + h²)
Surface Area = πr² + πr√ (r² + h²)
Step 4: Factor using disributive property
Divide all terms by πr:
Surface Area = πr (r + √ (r² + h²))
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*If you already have the slant height, use πrs. If you only have the radius and height, use the formula
above