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Advanced Geometry : Cones

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Example Question #31 : How To Find The Surface Area Of A Cone

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:

2509.88

2395.96

2261.95

1909.34

Correct answer:

2261.95

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.

$\text{Lateral Area}=(18)(12)\pi=216\pi$

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:

$\text{Surface Area of Cylindrical Portion}=2\pi(12)(15)+\pi(12)^2=504\pi$

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.

$\text{Surface Area of Figure}=216\pi+504\pi=720\pi=2261.95$

Make sure to round to places after the decimal.

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Example Question #31 : How To Find The Surface Area Of A Cone

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:

1790.71

2010.77

1851.25

1669.02

Correct answer:

1790.71

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.

$\text{Lateral Area}=(17)(10)\pi=170\pi$

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:

$\text{Surface Area of Cylindrical Portion}=2\pi(10)(15)+(10)^2\pi=400\pi$

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.

$\text{Surface Area of Figure}=170\pi+400\pi=570\pi=1790.71$

Make sure to round to places after the decimal.

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Example Question #31 : How To Find The Surface Area Of A Cone

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:

6918.71

7277.90

7409.57

8917.13

Correct answer:

7277.90

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.

$\text{Lateral Area}=(25\sqrt3)(22)\pi=(550\sqrt3)\pi$

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:

$\text{Surface Area of Cylindrical Portion}=2\pi(22)(20)+\pi(22)^2=1364\pi$

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.

$\text{Surface Area of Figure}=(550\sqrt3)\pi+1364\pi=7277.90$

Make sure to round to places after the decimal.

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Example Question #31 : Solid 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:

2077.08

2035.75

2166.39

2635.29

Correct answer:

2035.75

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.

$\text{Lateral Area}=(16)(12)\pi=192\pi$

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:

$\text{Surface Area of Cylindrical Portion}=2\pi(12)(13)+\pi(12)^2=456\pi$

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.

$\text{Surface Area of Figure}=192\pi+456\pi=648\pi=2035.75$

Make sure to round to places after the decimal.

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Example Question #31 : How To Find The Surface Area Of A Cone

Cone

Figure NOT drawn to scale.

The base of the above cone has circumference 50. Give the surface area of the cone.

Round to the nearest whole number.

Possible Answers:

306

301

737

712

699

Correct answer:

699

Explanation:

The surface area of a cone can be calculated using the formula

$S = \pi r (r+l)$

where is the radius of the base and is the slant height. The slant height is known to be 20.

The radius of the circular base can be found by dividing its circumference 50 by $2 \pi$ :

$r = \frac{50}{2 \pi} \approx \frac{50}{2 \cdot 3.1416} \approx 7.958$

Set r = 7.958 and l = 20 , and evaluate:

$S = \pi r (r+l)$

$S \approx 3.1416 \cdot 7.958 \cdot (7.958 + 20)$

$\approx 3.1416 \cdot 7.958 \cdot 27.958$

$\approx 699$

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Example Question #32 : How To Find The Surface Area Of A Cone

Cone

Figure NOT drawn to scale.

The base of the above cone has area 50. Give the surface area of the cone.

Round to the nearest whole number.

Possible Answers:

737

699

306

712

301

Correct answer:

301

Explanation:

The area and the radius of the circular base are related by the formula

$\pi r^{2}= A$

The area of the circle is 50, so set A = 50 , then solve for :

$\pi r^{2}= 50$

Divide by $\pi$ :

$\frac{\pi r^{2}}{\pi}= \frac{50}{\pi}$

$r^{2} \approx \frac{50}{3.1416}$

$r^{2} \approx 15.9155$

Take the square root of both sides:

$r \approx \sqrt{15.9155}$

$r \approx 3.9894$

The surface area of the cone is the sum of the areas of the base and the lateral area. The lateral area of the cone can be found from the radius and the slant height using the formula

$L = \pi r l$

Set $r \approx 3.9894$ and l = 20 :

$L \approx 3.1416 \cdot 3.9894 \cdot 20$

$L \approx 251$

Add the area of the base, 50:

$S \approx 251 + 50 = 301$

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Example Question #31 : Solid Geometry

Cone

Figure NOT drawn to scale.

The base of the above cone has area 50. Give the surface area of the cone.

Round to the nearest whole number.

Possible Answers:

699

737

306

301

712

Correct answer:

306

Explanation:

The area and the radius of the circular base are related by the formula

$\pi r^{2}= A$

The area of the circle is 50, so set A = 50 , then solve for :

$\pi r^{2}= 50$

Divide by $\pi$ :

$\frac{\pi r^{2}}{\pi}= \frac{50}{\pi}$

$r^{2} \approx \frac{50}{3.1416}$

$r^{2} \approx 15.9155$

Take the square root of both sides:

$r \approx \sqrt{15.9155}$

$r \approx 3.9894$

Now examine the figure below.

Cone

The slant height can be calculated from the height and the radius by way of the Pythagorean Theorem:

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

$l = \sqrt{20^{2}+ 3.9894^{2}}$

$\approx \sqrt{400 + 15.9153 }$

$\approx \sqrt{4 15.9153 }$

$\approx 20.3940$

The surface area of the cone is the sum of the areas of the base and the lateral area. The lateral area of the cone can be found from the radius and the slant height using the formula

$L = \pi r l$

Set $r \approx 3.9894$ and l = 20.3940 :

$L \approx 3.1416 \cdot 3.9894 \cdot 20.3940$

$L \approx 256$

Add the area of the base, 50, to this lateral area to obtain the surface area:

$S \approx 256 + 50 = 306$ .

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Example Question #33 : How To Find The Surface Area Of A Cone

Cone

Figure NOT drawn to scale.

The base of the above cone has circumference 50. Give the surface area of the cone.

Round to the nearest whole number.

Possible Answers:

699

301

737

306

712

Correct answer:

737

Explanation:

The surface area of a cone can be calculated using the formula

$S = \pi r (r+l)$

where is the radius of the base and is the slant height.

The radius of the circular base can be found by dividing its circumference 50 by $2 \pi$ :

$r = \frac{50}{2 \pi} \approx \frac{50}{2 \cdot 3.1416} \approx 7.958$

Examine the figure below.

Cone

The slant height can be calculated from the height and the radius by way of the Pythagorean Theorem:

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

$l = \sqrt{20^{2}+ 7.958^{2}}$

$\approx \sqrt{400 + 63.3298}$

$\approx \sqrt{4 63.3298}$

$\approx 21.5251$

Set $r \approx 7.958$ and $l \approx 21.5251$ in the surface area formula:

$S = \pi r (r+l)$

$S \approx 3.1416 \cdot 7.958 \cdot ( 7.958+21.5251 )$

$\approx 3.1416 \cdot 7.958 \cdot 29.4831$

$\approx 737$

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Advanced Geometry : Cones

Study concepts, example questions & explanations for Advanced Geometry

All Advanced Geometry Resources

Example Questions

Example Question #31 : How To Find The Surface Area Of A Cone

Example Question #31 : How To Find The Surface Area Of A Cone

Example Question #31 : How To Find The Surface Area Of A Cone

Example Question #31 : Solid Geometry

Example Question #31 : How To Find The Surface Area Of A Cone

Example Question #32 : How To Find The Surface Area Of A Cone

Example Question #31 : Solid Geometry

Example Question #33 : How To Find The Surface Area Of A Cone

All Advanced Geometry Resources

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