How to find the surface area of a cone - ACT Math

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Question

What is the surface area of a cone with a radius of 6 in and a height of 8 in?

Answer

Find the slant height of the cone using the Pythagorean theorem: _r_2 + _h_2 = _s_2 resulting in 62 + 82 = _s_2 leading to _s_2 = 100 or s = 10 in

SA = πrs + πr_2 = π(6)(10) + π(6)2 = 60_π + 36_π_ = 96_π_ in2

60_π_ in2 is the area of the cone without the base.

36_π_ in2 is the area of the base only.

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Question

Use the following formula to answer the question.

$SA=\pi r^2+\pi rl$

The slant height of a right circular cone is . The radius is , and the height is . Determine the surface area of the cone.

Answer

Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is .

First plug these numbers into the equation provided.

$SA=\pi r^2+\pi rl=\pi (2)^2+\pi (2)(6)=4\pi cm^2+12\pi cm^2$

Then simplify by combining like terms.

$4\pi cm^2+12\pi cm^2=16\pi cm^2$

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Question

The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of .

Answer

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 $\frac{1}{5}L$ ; the radius is half this, so

$r = \frac{1}{2} \cdot \frac{1}{5}L= \frac{1}{10}L$

Substitute in the surface area formula:

$A = \pi \left ( \frac{1}{10}L \right ) ^{2} + \pi \left ( \frac{1}{10}L \right ) L$

$A = \frac{1}{100} \pi L^{2} + \frac{1}{10} \pi L^{2}$

$A = \frac{1}{100} \pi L^{2} + \frac{10}{100} \pi L^{2}$

$A = \frac{11}{100} \pi L^{2} = \frac{11\pi L^{2}}{100}$

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Question

The height of a cone is ; the diameter of its base is twice the height. Give its surface area in terms of .

Answer

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 twice the height, which is ; the radius is half this, which is .

The slant height can be calculated using the Pythagorean Theorem:

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

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

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

$l =h \sqrt{2}$

Substitute $h \sqrt{2}$ for and for in the surface area formula:

$A = \pi h^{2} + \pi h \cdot h \sqrt{2}$

$A = \pi h^{2} + \pi h ^{2} \sqrt{2}$

$A = \left (1 + \sqrt{2} \right ) \pi h^{2}$

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Question

The radius of the base of a cone is ; its height is twice of the diameter of that base. Give its surface area in terms of .

Answer

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 base has radius and diameter . The height is twice the diamter, which is . Its slant height can be calculated using the Pythagorean Theorem:

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

$l = \sqrt{r^{2}+ (4r)^{2}}$

$l = \sqrt{r^{2}+ 16r^{2}}$

$l = \sqrt{17r^{2}}$

$l = r \sqrt{17 }$

Substitute $r \sqrt{17 }$ for in the surface area formula:

$A = \pi r^{2} + \pi r \cdot r \sqrt{17}$

$A = \pi r^{2} + \pi r^{2} \sqrt{17}$

$A = \left ( 1+ \sqrt{17} \right ) \pi r^{2}$

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Question

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).

Answer

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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Question

The circumference of the base of a cone is 80; the slant height of the cone is equal to twice the diameter of the base. Give the surface area of the cone (nearest whole number).

Answer

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 slant height is twice the diameter, or, equivalently, four times the radius, so

and

$A = \pi r^{2} + \pi r \cdot 4r$

$A = \pi r^{2} +4 \pi r^{2}$

$A = 5 \pi r^{2}$

The radius of the base is the circumference divided by $2\pi$ , which is

$80 \div 2 \pi \approx 80 \div (2 \times 3.14 )\approx 12.73$

Substitute:

$A = 5 \pi r^{2}$

$A \approx 5 \cdot 3.14 \cdot (12.73)^{2}$

$A \approx 2,546$

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Question

The radius of the base of a cone is ; its slant height is two-thirds of the diameter of that base. Give its surface area in terms of .

Answer

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 twice radius , or , and its slant height is two-thirds of this diameter, which is $\frac{2}{3}\cdot 2r = \frac{4}{3}r$ . Substitute this for in the formula:

$A = \pi r^{2} + \pi r \cdot \frac{4}{3}r$

$A = \pi r^{2} + \frac{4}{3} \pi r ^{2}$

$A = \frac{7}{3} \pi r ^{2} = \frac{7\pi r ^{2}}{3}$

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Question

Some conical party hats are being decorated with glitter paper. If the base of the hats is inches across and the slant height is inches, how many square inches of glitter paper are needed to decorate one hat?

Answer

The formula for the surface area of a cone is

$\pi r^2 + \pi rl$ , where is the slant height. Since we're only concerned with the slant height portion of the surface area formula, we can ignore the $\pi r^2$ portion of the equation.

Plug in known values to this part of the equation and solve.

$\pi rl = \pi (6)(8) = 48\pi$

So, each hat requires $48\pi \textup{ in}^2$ of glitter paper.

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Question

A heat shield on a particular satellite takes the form of a cone. If the surface area of the "face" of the cone (not counting the disk on the bottom) is , and the stripe of reflective paint from the tip of the cone is down to the base is feet long, what is the diameter of the disk in feet? Round $\pi$ to 3 significant digits. Round your final answer to the nearest foot.

Answer

In this problem, we only need to consider the part of the formula for conic surface area that deals with slant height, since that is all the information we have.

We know the formula for the lateral surface area of a cone is $\pi r l$ , and we know that is feet. Plugging in our other values gives us:

$\pi r (38) = 2744.36$

Simplify:

$\pi r (38) = 2744.36$

$r = \frac{2744.36}{(38)(3.14)} \approx 23$

Thus, if our radius is approximately feet, our diameter is approximately feet.

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Question

What is the surface area in square units of a cone with radius units and height units?

Answer

The formula for surface area of a cone is:

$\pi r^2 + \pi r l$ , where is the slant height. Since we know the radius, we can calculate the first part without issue:

$\pi r^2 = \pi (7)^2 = 49\pi$

The second part requires us to calculate slant height. Since all cones have a right angle created by the base and height perpendicular to the base, we can use the Pythagorean theorem to calculate :

Now, we can complete our formula. Don't forget to add in the circular base.

$\pi r^2 + \pi r l = 49\pi + \pi (7)(25) = 49\pi + 175\pi = 224\pi$

Thus, our surface area is $224\pi$ square units.

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Question

The surface area of a cone equals $24\pi \textup{cm}^2.$ If the radius of the cone equals $3\textup{cm}$ , what is the height of the cone?

Answer

To solve this question, you need to know the surface area of a cone equation: $\textup{surface area}=\pi r(r+\sqrt{h^2 +r^2}),$ where is the radius of the cone $(3\textup{cm})$ , and is the height of the cone. We must substitute all of the values that we know, and solve for the height of the cone to get the answer.

$24\textup{cm}^2=\pi *3\textup{cm}(3\textup{cm}+\sqrt{h^2 +(3\textup{cm})^2}),$ Therefore, the height of the cone is $4\textup{cm.}$

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