The given solid is a with a of 4.5 centimeters and a height of 12 centimeters.
To calculate the of a cone, we can use the known formula where
r is the of the base and
ℓ is the .
S=πrℓ+πr2
Looking at the diagram, we know that the diameter of the cone's base is
4.5 centimeters. By dividing by
2, we get the radius of the base.
r=24.5 ⇔ r=2.25 cm
We now know that the radius of the cone is
2.25 centimeters and that the height of the cone is
12 centimeters. Now, we can focus on the created within the cone.
The slant height of the cone is the of the triangle. The legs of the triangle are the height of the pyramid
h and the radius
r of the base. Notice that we can find the slant height of the cone using the .
h2+r2=ℓ2
Let's substitute
12 for
h and
2.25 for
r into the above equation to find the slant height of the cone.
h2+r2=ℓ2
(12)2+(2.25)2=ℓ2
144+5.0625=ℓ2
149.0625=ℓ2
ℓ2=149.0625
ℓ=12.209115
ℓ≈12.2
We found that the slant height is about
12.2 centimeters. Note that, when solving the above equation, we only kept the because
ℓ represents a side length and must be a number. Let's now substitute
r with
2.25 and
ℓ with
12.2 into the formula, we can calculate
S.
S=πrℓ+πr2
S=π(2.25)(12.2)+π(2.25)2
S=π(2.25)(12.2)+5.0625π
S=27.45π+5.0625π
S=32.5125π
S=102.141031…
S≈102.1
The surface area of the cone is approximately
102.1 square centimeters.