How to find the surface area of a cone - High School Math

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Question

The lateral area is twice as big as the base area of a cone. If the height of the cone is 9, what is the entire surface area (base area plus lateral area)?

Answer

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r2)

rl = 2r2

l = 2r

From the diagram, we can see that r2 + h2 = l2. Since h = 9 and l = 2r, some substitution yields

r2 + 92 = (2r)2

r2 + 81 = 4r2

81 = 3r2

27 = r2

B = π(r2) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

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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

What is the surface area of a cone with a radius of 4 and a height of 3?

Answer

Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

$\Pi r^{2} + \Pi r\sqrt{r^{2} + h^{2}}= \Pi\ast 4^{2} + \Pi \ast 4\sqrt{4^{2} + 3^{2}} = 16\Pi + 4\Pi \sqrt{25} = 16\Pi + 20\Pi = 36\Pi$

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Question

What is the surface area of a cone with a height of 8 and a base with a radius of 5?

Answer

To find the surface area of a cone we must plug in the appropriate numbers into the equation

$Surface\ Area=\pi r^{2}+\pi rl$

where is the radius of the base, and is the lateral, or slant height of the cone.

First we must find the area of the circle.

To find the area of the circle we plug in our radius into the equation of a circle which is

This yields $A=25\pi$ .

We then need to know the surface area of the cone shape.

To find this we must use our height and our radius to make a right triangle in order to find the lateral height using Pythagorean’s Theorem.

Pythagorean’s Theorem states

Take the radius and height and plug them into the equation as a and b to yield $5^{2}+8^{2}=c^{2}$

First square the numbers $25+64=c^{2}$

After squaring the numbers add them together $89=c^{2}$

Once you have the sum, square root both sides $\sqrt{89}=\sqrt{c^{2}}$

After calculating we find our length is

Then plug the length into the second portion of our surface area equation above to get $(4)(\pi)(9.43)=37.72\pi$

Then add the area of the circle with the conical area to find the surface area of the entire figure $25\pi+37.72\pi=62.72\pi$

The answer is $62.72\pi$ .

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Question

Find the surface area of a cone that has a radius of 12 and a slant height of 15.

Answer

The standard equation to find the surface area of a cone is

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

where denotes the slant height of the cone, and denotes the radius.

Plug in the given values for and to find the answer:

$SA=\pi (12)(15)+\pi (12)^2=180\pi +144\pi =324\pi$

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Question

Find the surface area of the following cone.

Answer

The formula for the surface area of a cone is:

$SA = A_{base} + A_{lateral}$

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

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

Plugging in our values, we get:

$SA = \pi (5m)^2 + \pi (5m) (13m)$

$SA = 25 \pi m^2 + 65 \pi m^2 = 90 \pi m^2$

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Question

Find the surface area of the following cone.

Answer

The formula for the surface area of a cone is:

$SA = (base) + \frac{(circumference)(slant height)}{2}$

$SA = (\pi r^2) + \frac{(2 \pi r)(h_s)}{2}$

$SA = (\pi r^2) + ( \pi rh_s)$

Use the Pythagorean Theorem to find the length of the radius:

Plugging in our values, we get:

$SA = \pi (6m)^2 + \pi (6m)(10m)$

$SA = 96 \pi m^2$

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Question

Find the surface area of the following half cone.

Answer

The formula for the surface area of the half cone is:

$SA=(base)+\frac{1}{2}(circumference)(slant\ height)+(triangle)$

$SA=\left(\frac{\pi r^2}{2}\right)+\left(\frac{1}{2}\cdot \frac{2\pi r h_s}{2}\right)+rh$

Where is the radius, is the slant height, and is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

Plugging in our values, we get:

$SA=\left(\frac{\pi (8m)^2}{2}\right)+\left( \frac{\pi (8m) (17m)}{2}\right)+(8m)(15m)$

$SA=32 \pi m^2 + 68 \pi m^2 + 120m^2$

$SA = 100 \pi m^2 + 120 m^2$

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