To find the radius of the base, use the fact that the arc length of the circular sector is equal to the circumference of the base. The area of a circular sector is one-half the radius multiplied by the arc length.
See solution.
Practice makes perfect
In Exploration 1 we drew a circle with radius 3 inches, divided it into 6 equal parts, and then cut one sector of the circle to make a cone.
The arc length of the sector above is 5Ď€, which will be the circumference of the base, C=5Ď€. On the other hand, we also know that C=2Ď€ r, where r is the radius of the base.
Thus, the radius of the base is r=2.5 inches. To find the surface area, we must add the area of the base and the lateral area.
S = A_(base) + A_(lateral)
The area of the base is π(2.5)^2 and the lateral area is the area of the sector, which is one-half the radius by the arc length.
A_(lateral) = 1/2* 3 * 5Ď€ = 7.5Ď€
Let's substitute and add the two corresponding areas.
In consequence, the surface area of the cone is 13.75Ď€ in^2.
Cutting Two Sectors
In this part, let's cut two sectors of the circle and make a second cone.
As before, to find the radius of the base we divide the arc length of the sector by 2Ď€.
r = 4Ď€/2Ď€=2
The lateral area of this new cone is the area of the sector, which we find as we did above.
A_(lateral) = 1/2* 3 * 4Ď€ = 6Ď€
Finally, we find the surface area of the second cone by adding the area of the base and the lateral area.
Once more, we repeat the process we did above, but this time we will cut three sectors.
The radius of the base is the quotient between the arc length of the sector and 2Ď€.
r = 3Ď€/2Ď€=1.5
The lateral area of this cone is the area of the sector, which we find as we did before.
A_(lateral) = 1/2* 3 * 3Ď€ = 4.5Ď€
The surface area of the third cone is the sum between the area of the base and the lateral area.
The surface area of the second cone is 6.75Ď€ in^2.
Cutting Four Sectors
One last time we repeat the process, but this time we cut four sectors.
To find the radius of the base, we divide the arc length of the sector by 2Ď€.
r = 2Ď€/2Ď€=1
The lateral area of this new cone is the area of the sector.
A_(lateral) = 1/2* 3 * 2Ď€ = 3Ď€
Finally, we find the surface area of the second cone by adding the area of the base and the lateral area.
In the following table, we will summarize all the data we found before.
# of sectors cut
Radius of the Base (in)
Surface Area (in^2)
1
5Ď€/2Ď€ = 2.5
13.75Ď€
2
4Ď€/2Ď€ = 2
10Ď€
3
3Ď€/2Ď€ = 1.5
6.75Ď€
4
2Ď€/2Ď€ = 1
4Ď€
From the table, we can see that the radius of the base is decreasing 0.5 inches every time. The surface area first decreased 3.75Ď€ in^2, then 3.25Ď€ in^2, and then 2.75Ď€ in^2.
# of sectors cut
Radius of the Base (in)
Surface Area (in^2)
1
5Ď€/2Ď€ = 2.5
13.75Ď€
2
4π/2π = 2 ↓ -0.5
10π ↓ -3.75π
3
3π/2π = 1.5 ↓ -0.5
6.75π ↓ -3.25π
4
2π/2π = 1 ↓ -0.5
4π ↓ -2.75π
Extra
Extra
From the table, we can see that if we cut five sectors, the radius of the base of the resulting cone will be r=0.5 inches and the surface area will be 4Ď€ - 2.25Ď€ = 1.75Ď€ in^2.
