Surface Area of Cone

Let's say you wanted to work out the surface area of an ice cream cone. There are a few things you might want to know before you can start, such as "why do you want to work out the surface area of an ice cream cone?" or, after you've had that conversation, "how do we calculate the surface area of the cone?". To answer that question, you'll need the formula for the surface area of a cone, the radius, and the slant length of the ice cream cone. So that's what we're going to cover here.

Surface Area of Cone Surface Area of Cone

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Contents
Table of contents

    What is the surface area of a cone?

    The surface area of a cone is the total surface area covered by both of its sides, so the sum of the area of its circular base and its curved surface.

    You should try imagining what a cone looks like, think of the body or the sides of a cone. This would give you an idea of the task.

    Which of the following objects is most likely to have a conical surface - a ball, a funnel, a plate, or a bed?

    Solution:

    From the list of items, only a funnel has a conical surface.

    Curved surface area of a cone

    The curved surface area of a cone is the area of the cone's body without the base. Here the slant height of the cone is very important.

    Surface of Cones, Illustrating the curved surface area of a cone, StudySmarterIllustrating the curved surface area of a cone, StudySmarter Originals

    Calculating the curved surface area of a cone

    The curved surface area of a cone is calculated by multiplying pi, the radius and slant height of a cone.

    Hence, the curved surface area of a cone, \(A_{cs}\) is given as:

    \[A_{cs}=\pi rl\]

    where \(r\) is the radius of the circular base of the cone, and \(l\) is the slant height of the cone.

    Find the curved surface area of a cone with radius \(7\, cm\) and slant height \(10\, cm\). Take \(\pi=\frac{22}{7}\)

    Solution:

    Since pi, radius, and slant height have been given, you should apply the formula. Hence the curved surface area of the cone is calculated as

    \[A_{cs}=\frac{22}{7}\times 7\, cm \times 10\, cm\]

    \[A_{cs}=220\, cm^2\]

    Surface area of a cone formula

    As stated before, the surface area of a cone is the total combined surface area of its curved surface and circular base, so we can make some logical assumptions as to what the formula might be, but we'll go into the derivation of the formula soon. Here, however, is the formula you must know:

    a=πr2+πrl

    In this case, "a" is the total surface area, "r" is the radius of the circular base and "l" is the length of the curved surface (usually called slant height). l is not the internal height, they are two different measurements. The image below shows this in the case of a cone, to give you a better understanding.

    Surface of Cones, A labelled diagram of a cone, StudySmarterA labelled diagram of a cone, StudySmarter Originals

    If you are given the internal height of a cone, you can use the Pythagorean theorem to calculate the slant length.

    Surface of Cones, An illustration on how the slant height is derived from the radius and the height, StudySmarterAn illustration on how the slant height is derived from the radius and the height, StudySmarter Originals

    Surface area of cone derivation

    Now that we know the formula, we should talk about how we can derive it from some other bits of information. Assuming we split the side (slant height side) of a cone and spread it out, we have what is displayed in the diagram below.

    The main thing we need to remember is that a cone can be broken down into two sections, the circular base and the conical section or curved surface.

    Surface of Cones, An illustration on the derivation of the total surface area of a cone, StudySmarterAn illustration on the derivation of the total surface area of a cone, StudySmarter Originals

    1. Separate the curved surface and the circular base. It's easier for you to calculate the surface area of each part separately. Forget about the circle section, for now, you will come back to it.
    2. If you take the conical section and unfold it, you will see that it is actually a sector of a larger circle that has a radius of l. The circumference of this larger circle is therefore2πland the area isπl2. The length of the arc of the sector you have is the same length as the circumference of the original circle section, which is2πr.
    3. The ratio between the area of the whole circle and the ratio of the area of the sector is the same as the ratio between the entire circumference and the part of the circumference of the sector. If you take the area of the sector to be "a", you can put this into an equation: \[\frac{a}{whole\, circle\, area}=\frac{arc\, length}{whole\, circle\, circumference}\]

    4. We substitute the values from step 2 into the word equation from step 3: aπl2=2πr2πl
    5. In this step, we're just going to look at what we need to do to simplify the above equation.

      The2π on the right-hand side both cancel:

      aπl2=2πr2πl

      Then we multiply both sides by πl2:

      a=rlπl2

      This allows us to cancel out some l's:

      a=rlπl2

      And that leaves us with:

      a=πrl

    6. Remember our circle from earlier? Well, the area of a circle is πr2 and the area of our conical section is πrl, so if we take both of these areas and combine them we get the total surface area of a cone, which is:

    a=πr2+πrl

    Finding the surface area of a cone

    Given a cone with a base radius of 7 feet and an internal height of 12 feet, calculate the surface area.

    Solution:

    As we have been given the internal height, we need to use Pythagoras' theorem to calculate the slant height:

    72 + 122 = 193

    Slant height =193

    We can take the formula and see what numbers we can plug into it: a=πr2+πrl

    7 is our radius r, and 193 is our slant height l.

    a=(π×72)+(π×7×193)

    a=49π+305.511

    a=459.45

    So our final answer, in this case, would be that a = 459.45 ft2, as the area is measured in units2.

    Given a cone with a base diameter of 14 feet and an internal height of 18 feet, calculate the surface area.

    Solution:

    We need to be careful in this case, as we've been given the bottom length as diameter and not a radius. The radius is simply half of the diameter, so the radius in this case is 7 feet. Again, we need to use the Pythagorean theorem to calculate the slant height:

    182 + 72 = 373

    Slant height = 373

    We take the formula and then substitute r for 7 and l for 373:

    a=(π×72)+(π×7×373)

    a=49π+424.720

    a=578.66

    Therefore, our final answer is a = 578.66 ft2

    Examples of surface of cones

    In order to improve your ability in solving questions on surface of cones, you are advised to practice more problems.

    From the figure below find the curved surface area of the cone.

    Surface of Cones, Examples of curved surface are without the slant height, StudySmarterExamples of curved surface are without the slant height, StudySmarter Originals

    Take \(\pi=3.14\)

    Solution:

    In this problem, you have been given the radius and height but not the slant height.

    Recall that the height of a cone is perpendicular to the radius so that with the slant height, a right-angle triangle is formed.

    Surface of Cones, Deriving the slant height of a cone when not given, StudySmarterDeriving the slant height of a cone when not given, StudySmarter Originals

    By using Pythagoras theorem,

    \[l=\sqrt{8^2+3.5^2}\]

    \[l=8.73\, m\]

    Now you can find the curved surface area

    Use \(A_{cs}=\pi rl\). I hope you didn't forget

    \[A_{cs}=3.14\times 3.5\, m \times 8.73\, m\]

    Thus, the curved surface area of the cone, \(A_{cs}\) is:

    \[A_{cs}=95.94\, m^2\]

    In Ikeduru palm fruits are arranged in a conical manner, they are required to be covered with palm fronds of average area \(6\, m^2\) and mass \(10\, kg\). If the palm is inclined at an angle \(30°\) to the horizontal, and the base distance of a conical stock of palm fruits is \(100\, m\). Find the mass of palm frond needed to cover the stock of palm fruits. Take \(\pi=3.14\).

    Solution:

    Make a sketch of the story.

    Is that a story or a question? Not sure, just solve it

    Surface of Cones, Finding the area of a cone with a given angle, StudySmarterFinding the area of a cone with a given angle, StudySmarter Originals

    So you can use SOHCAHTOA to get your slant height since

    \[\cos\theta=\frac{adjacent}{hypotenuse}\]

    The \(50\, m\) was gotten from halving the base distance since we need the radius.

    \[\cos(30°)=\frac{50\, m}{l}\]

    Cross multiply

    Note that \[\cos(30°)=0.866\]

    \[0.866l=50\, m\]

    Divide both sides by \(0.866\) to get the slant height, \(l\)

    \[l=57.74\, m\]

    Now you can find the total surface area of the conical stock knowing that

    \[a=\pi r^2+\pi rl\]

    Hence

    \[a=(3.14\times (50\, m)^2)+(3.14\times 50\, m \times 57.74\, m)\]

    \[a=7850\, m^2+9065.18\, m^2\]

    Hence, the area of the conical stock is \(16915.18\, m^2\).

    However, your task is to know the weight of palm fronds used to cover the conical stock. To do this, you need to know how many palm fronds would cover the stock since the area of a palm frond is \(6\, m^2\). Thus the number of palm fronds required, \(N_{pf}\) is

    \[N_{pf}=\frac{16915.18\, m^2}{6\, m^2}\]

    \[N_{pf}=2819.2\, fronds\]

    With each palm frond weighing \(10\, kg\), the total mass of frond needed to cover the conical palm fruit stock, \(M_{pf}\) is:

    \[M_{pf}=2819.2 \times 10\, kg\]

    \[M_{pf}=28192\, kg\]

    Therefore the mass of palm frond required to cover an average conical stock of palm fruit in Ikeduru is \(28192\, kg\).

    Surface of Cones - Key takeaways

    • The surface area of a cone is the sum of the surface area of the circular base and the conical section.
    • The formula for calculating the surface area of a cone is a=πr2+πrl where r is the radius of the circle at the base and l is the height of the slant.
    • If you are asked for the surface area of a cone but are given internal height instead of slant height, use Pythagoras' theorem to calculate the slant height.
    Frequently Asked Questions about Surface Area of Cone

    What is the surface area of a cone?

    The surface area of a cone is the total surface area covered by both of its sides, so the sum of the area of its circular base and its curved surface. 

    What is the formula for the surface of a cone?

    a = πr2+πrl

    How to derive the surface area of a cone?

    To determine the surface area of cone derivation, we cut the cone open from the center which looks like a sector of a circle. Now what we have depicts;

    The total surface area of cone = area of the base of cone + curved surface area of a cone

    How to calculate the surface area of a cone without base?

    Use the formula;

    Area of the curved surface= πrl 

    What is the equation for the surface area of a cone?

    The equation for the surface area of a cone is the same as the formula used in calculating the total surface area of a cone which is: a = πr2+πrl

    Test your knowledge with multiple choice flashcards

    Given a cone of internal height of 48 feet and base diameter of 28 feet, what is the surface area?

    Given a cone with a radius of 3 feet and a slant height of 7.6 feet, calculate the surface area of the cone.

    A cone has a slant height of 10 inches and a radius of 7 inches, what is the internal height?

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