Área de superficie de un cilindro: explicación y ejemplos

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Pau Monfort
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Área de superficie de un cilindro: explicación y ejemplos

Before we jump into the topic of a cylinder’s surface area, let’s review a cylinder. In geometry, a cylinder is a three-dimensional figure with two circular bases parallel to each other and a curved surface.


How to Find the Surface Area of a Cylinder?

The surface area of a cylinder is the sum of two parallel and congruent circular faces and the curved surface area.


Este artículo discutirá how to find the total surface area and lateral surface area of a cylinder.

To calculate the surface area of a cylinder, you need to find the Base Area (B) and Curved Surface Area (CSA). Therefore, the surface area or the total surface of a cylinder is equal to the sum of the base area times two and the area of the curved surface.

The curved surface of a cylinder is equal to a rectangle whose length is 2πr and whose width is h.

Where r = radius of the circular face and h = height of the cylinder.

The area of the curved surface = Area of a rectangle =l x w = πdh


The base area, B = Area of a circle = πr2

The area of a cylinder formula

The formula for the total surface area of a cylinder is given as:

Total surface area of a cylinder = 2πr2 + 2πrh

TSA = 2πr2 + 2πrh

Where 2πr2 is the top and bottom circular face area, and 2πrh is the area of the curved surface.


By taking 2πr as a common factor from RHS, we get;

TSA = 2πr (h + r) ……………………………………. (Surface area of a cylinder formula)

Let’s solve example problems involving the surface area of a cylinder.

ejemplo 1

Find the total surface area of a cylinder whose radius is 5 cm and height is 7 cm.

Solución

Por la fórmula,

TSA = 2πr (h + r)

= 2 x 3.14 x 5(7 + 5)

= 31.4 x 12

= 376.8 cm2

ejemplo 2

Find the radius of a cylinder whose total surface area is 2136.56 square feet, and the height is 3 feet.

Solución

Dado:

TSA = 2136.56 square feet

Height, h = 3 feet

But, TSA = 2πr (h + r)

2136.56 =2 x 3.14 x r (3 + r)

2136.56 = 6.28r (3 + r)

By distributive property of multiplication on the RHS, we have,


2136.56 = 18.84r + 6.28r2

Dividir cada término por 6.28

340.22 = 3r + r2

r2 + 3r – 340.22 = 0 ……… (a quadratic equation)

By solving the equation using the quadratic formula, we get,

r = 17

Therefore, the radius of the cylinder is 17 feet.

ejemplo 3

The cost of painting a cylindrical container is $0.04 per cm2. Find the cost of painting 20 containers of radius, 50 cm, and height, 80 cm.

Solución

Calculate the total surface area of 20 containers.


TSA = 2πr (h + r)

= 2 x 3.14 x 50 (80 + 50)

= 314 x 130

= 40820 cm2

The total surface area of 20 containers = 40,820 cm2 x 20

= 816,400 cm2

The cost of painting = 816,400 cm2   x $0.04 per cm2


= $ 32,656.

Hence, the cost of painting 20 containers is $32,656.

ejemplo 4

Find the height of a cylinder if its total surface area is 2552 in2 and the radius is 14 in.

Solución

Dado:

TSA = 2552 pulg2

Radius, r = 14 in.

But, TSA = 2πr (h + r)

2552 = 2 x 3.14 x 14 (14 + h)

2552 = 87.92(14 + h)

Divide ambos lados entre 87.92 para obtener,

29.026 = 14 + h

Resta 14 en ambos lados.

h = 15

Hence, the height of the cylinder is 15 in.

Lateral Surface Area of a Cylinder

As stated before, the area of the curved surface of a cylinder is what is termed as the lateral surface area. In simple words, a cylinder’s lateral surface area is the surface area of a cylinder, excluding the area of the base and bottom (circular surface).


The formula gives the lateral surface area of a cylinder;

LSA = 2πrh

ejemplo 5

Find the later surface area of a cylinder whose diameter is 56 cm and height is 20 cm.

Solución

Dado:

Diameter = 56 cm, hence radius, r =56/2 = 28 cm

Altura, h = 20 cm

By, the formula,

LSA = 2πrh

= 2 x 3.14 x 28 x 20

= 3516.8 cm2.

Thus, the lateral surface area of the cylinder is 3516.8 cm2.

ejemplo 6

The lateral surface area of a cylinder is 144 ft2. If the radius of the cylinder is 7 ft, find the height of the cylinder.

Solución

Dado;

LSA = 144 ft2

Radius, r = 7 ft

144 = 2 x 3.14 x 7 x h

144 = 43.96h

Divide by 43.96 on both sides.

3.28 = h

So, the height of the cylinder is 3.28 ft.



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