The formula for the surface area of a cone with base of radius \(\displaystyle r\) and slant height \(\displaystyle l\) is

\(\displaystyle A = \pi r^{2} + \pi r l\).

The diameter of the base is the circumference divided by \(\displaystyle \pi\), which is 

\(\displaystyle d = C \div (2\pi ) \approx 100 \div 3.14 \approx 31.83\)

This is also the height \(\displaystyle h\).

The radius is half this, or 

\(\displaystyle r = d \div 2 \approx 31.83 \div 2 \approx 15.92\)

The slant height can be found by way of the Pythagorean Theorem:

\(\displaystyle l = \sqrt{r^{2}+ h ^{2}}\)

\(\displaystyle l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}\)

\(\displaystyle l \approx \sqrt{253.30+ 1,013.21}\)

\(\displaystyle l \approx \sqrt{1266.51}\)

\(\displaystyle l \approx 35.60\)

Substitute in the surface area formula:

\(\displaystyle A = \pi r^{2} + \pi r l\)

\(\displaystyle A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60\)

\(\displaystyle A \approx 795.8+ 1,780.1\)

\(\displaystyle A \approx 2,575\)

When creating a net image of a 3D figure - one imagines it is made of paper and is unfurled into its' 2D form. The lateral portion of the cone cone would be unfurled into the image of a Sector of a Circle. To include the full surface area of the cone a circle is included to form the base of the cone as in the figure below. The lateral area portion is the top part of the figure below. 

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(15)(14)\pi=210\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(14)(10)+\pi(14)^2=476\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=210\pi+476\pi=686\pi=2155.13\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(20)(12)\pi=240\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(12)(8)+\pi(12)^2=336\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=240\pi+336\pi=576\pi=1809.56\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(8)(4)\pi=32\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(4)(2)+\pi(4)^2=32\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=32\pi+32\pi=64\pi=201.06\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(13)(5)\pi=65\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(5)(9)+\pi(5)^2=115\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=65\pi+115\pi=180\pi=565.49\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(15)(4)\pi=60\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(4)(6)+\pi(4)^2=64\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=60\pi+64\pi=124\pi=389.56\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(9)(4)\pi=36\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(9)(4)+\pi(4)^2=88\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=36\pi+88\pi=124\pi=389.56\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(15)(10)\pi=150\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(10)(8)+\pi(10)^2=260\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=150\pi+260\pi=410\pi=1288.05\)

Make sure to round to \(\displaystyle 2\) places after the decimal.

First, find the lateral surface area of the cone.

\(\displaystyle \text{Lateral Surface Area}=(\text{slant height})(\pi)(\text{radius})\)

Plug in the given slant height and radius.

\(\displaystyle \text{Lateral Area}=(13)(10)\pi=130\pi\)

Next, find the surface area of the cylinder:

\(\displaystyle \text{Surface Area of Cylinder}=2\pi(\text{radius})(\text{height})+2\pi(\text{radius})^2\)

The surface area of the cylinder is the sum of the lateral surface area of the cylinder and its two bases. However, since one base of the cylinder is covered up by the cone, we will need to subtract that area out of the total surface area of the cylinder.

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(\text{radius})(\text{height})+\pi(\text{radius})^2\)

Plug in the given height and radius to find the surface area of the cylindrical portion of the figure:

\(\displaystyle \text{Surface Area of Cylindrical Portion}=2\pi(10)(12)+\pi(10)^2=340\pi\)

To find the surface area of the figure, add together the lateral area of the cone with the surface area of the cylindrical portion of the figure.

\(\displaystyle \text{Surface Area of Figure}=130\pi+340\pi=470\pi=1476.55\)

Make sure to round to \(\displaystyle 2\) places after the decimal.