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Advanced Geometry : How to find the surface area of a cone

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Example Questions

Example Question #21 : Advanced Geometry

The circumference of the base of a cone is 100; the height of the cone is equal to the diameter of the base. Give the surface area of the cone (nearest whole number).

Possible Answers:

2,575

2,385

1,589

1,779

8,443

Correct answer:

2,575

Explanation:

The formula for the surface area of a cone with base of radius and slant height is

$A = \pi r^{2} + \pi r l$ .

The diameter of the base is the circumference divided by $\pi$ , which is

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

This is also the height .

The radius is half this, or

$r = d \div 2 \approx 31.83 \div 2 \approx 15.92$

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

$l = \sqrt{r^{2}+ h ^{2}}$

$l \approx \sqrt{15.92^{2}+ 31.83 ^{2}}$

$l \approx \sqrt{253.30+ 1,013.21}$

$l \approx \sqrt{1266.51}$

$l \approx 35.60$

Substitute in the surface area formula:

$A = \pi r^{2} + \pi r l$

$A \approx 3.14 \cdot 15.92 ^{2} + 3.14 \cdot 15.92 \cdot 35.60$

$A \approx 795.8+ 1,780.1$

$A \approx 2,575$

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Example Question #21 : Advanced Geometry

If a cone were unfurled into a 2-dimensional figure. The lateral area of the cone would look most like which figure?

Possible Answers:

Sector of a Circle

Rectangle

Circle

Triangle

Correct answer:

Sector of a Circle

Explanation:

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.

Cone net

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Example Question #21 : How To Find The Surface Area Of A Cone

As shown by the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

2155.13

2274.11

2335.69

2635.09

Correct answer:

2155.13

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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.

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

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

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

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.

$\text{Surface Area of Figure}=210\pi+476\pi=686\pi=2155.13$

Make sure to round to places after the decimal.

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Example Question #23 : Cones

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

1966.37

1809.56

1745.68

1817.24

Correct answer:

1809.56

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=240\pi+336\pi=576\pi=1809.56$

Make sure to round to places after the decimal.

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Example Question #21 : Advanced Geometry

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

258.43

196.35

208.09

201.06

Correct answer:

201.06

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=32\pi+32\pi=64\pi=201.06$

Make sure to round to places after the decimal.

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Example Question #21 : Cones

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

652.22

581.74

565.49

509.31

Correct answer:

565.49

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=65\pi+115\pi=180\pi=565.49$

Make sure to round to places after the decimal.

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Example Question #26 : Cones

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

389.56

411.22

398.32

376.09

Correct answer:

389.56

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=60\pi+64\pi=124\pi=389.56$

Make sure to round to places after the decimal.

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Example Question #27 : Cones

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

410.58

353.08

389.56

374.19

Correct answer:

389.56

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=36\pi+88\pi=124\pi=389.56$

Make sure to round to places after the decimal.

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Example Question #28 : Cones

In the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

1288.05

1309.67

1244.32

1316.59

Correct answer:

1288.05

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=150\pi+260\pi=410\pi=1288.05$

Make sure to round to places after the decimal.

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Example Question #29 : Cones

As shown by the figure below, a cone is placed on top of a cylinder so that they share the same base. Find the surface area of the figure.

Possible Answers:

1476.55

1341.61

1509.20

1492.19

Correct answer:

1476.55

Explanation:

First, find the lateral surface area of the cone.

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

Plug in the given slant height and radius.

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

Next, find the surface area of the cylinder:

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

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

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

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

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.

$\text{Surface Area of Figure}=130\pi+340\pi=470\pi=1476.55$

Make sure to round to places after the decimal.

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Advanced Geometry : How to find the surface area of a cone

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #21 : Advanced Geometry

Example Question #21 : Advanced Geometry

Example Question #21 : How To Find The Surface Area Of A Cone

Example Question #23 : Cones

Example Question #21 : Advanced Geometry

Example Question #21 : Cones

Example Question #26 : Cones

Example Question #27 : Cones

Example Question #28 : Cones

Example Question #29 : Cones

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