Surface Area Of A Right Circular Cylinder
Mensuration of Class 10
The surface area of a cylinder can be defined as the total space covered by the flat surfaces of the base of the cylinder and its curved surface. The total surface of the cylinder has two components - a curved surface and two flat surfaces. Let's learn more about what the surface area of a cylinder is and how to calculate the total surface area and lateral surface area of a cylinder.
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What is the Surface Area of a Cylinder?
An area of a cylinder is the total area covered by a cylinder in three-dimensional space. The area of the cylinder is equal to the sum of the area of the two circular bases and the curved surface area. The two circular bases are exactly above each other in right cylinders,, and the axis makes a right angle to the base. If one of the bases of circular is displaced and the axis is not at the base at right angles, it is called an oblique cylinder.
In the middle of the two circular bases is a curved surface that presents a rectangular shape when opened. This curved surface is also called the lateral surface. The various parameters used to calculate a cylinder's area include the axis, base, radius, height, and side. The radius of the cylinder is known as the radius of the circular base. The cylinder height is calculated by measuring the perpendicular distance between the 2 circular bases, and the line that joins the center of the base is called the axis.
Curved Surface Area or Lateral Surface Area of Cylinder Formula
The curved surface of a cylinder is the surface covered only by its curved surface. If the radius of the base of the cylinder is "r" and the height of the cylinder is "h", the curved surface of the cylinder is calculated according to the following formula:
Formula of CSA of Cylinder is “2πrh”
where,
r = radius of the cylinder
h = height of cylinder
π = 22/7 or 3.14
Total Surface Area of Cylinder Formula
A total surface area of the cylinder is occur by sum of the area of the curved surface and area of the two bases. Therefore, the formula for the total surface area of the cylinder is given as,
Total area of cylinder = area of two bases + area of curved surface. Since the bases of the cylinder are in the shape of circular, their total area will be πr2 + πr2. We already know that the curved surface of a cylinder is 2πrh.
TSA of cylinder = (πr2 + πr2) + 2πrh
= 2πr2 + 2πrh
Derivation of the Formula for the Surface Area of Cylinder
An area of any shape is the space it occupies. A cylinder has 2 flat surfaces, that are circles, and a curved surface which opens up as a rectangle. Consider the cylinder below whose height is 'h' and radius is 'r.' Let's open the cylinder in 2-dimensional form and understand it.
Note the figure above in which the area of the curved surface opens up as a rectangle, and the two bases are circles. Now the area of two circles is (πr2 + πr2), whose base radius is 'r.' In a rectangle, h is the 1 side of the cylinder h, whereas the length of the given rectangle is the circumference of the circle, i.e., 2πr. An area of this rectangle (l × b) is, therefore, = 2πr × h = 2πrh, which is also the given curved surface area of the cylinder. So, the total surface area of the cylinder = 2πr2 + 2πrh = 2πr(r + h)
How to Calculate the Surface Area of a Cylinder?
The surface area of the cylinder is equal to the area that is occupied by the curved surface of the cylinder and the bases of the cylinder. Use the steps below to find a cylinder's total surface area with a radius of 5 units and a height of 8 units.
- Step 1: Check the radius "r" and height "h" of the cylinder. Make sure they both have the same units. Here r =5, h = 8
- Step 2: In the question, we have to find the total surface area of the cylinder, so we use the formula for the total surface area of a cylinder, the total surface area of the cylinder = 2πr(r + h)
- Step 3: Replace the given values and then give the answer in square units. After replacing the values in the formula, we get the total surface = 2πr(r + h)= 2π × 5(5 + 8) ⇒ 2π ×45 ⇒ 2 × 3.14 × 45 ⇒ 2826 square units.
Solved Examples
Q1. The radius of a cylinder and the height of the cylinder are 4 inches and 14 inches. Find the surface area of the cylinder.
Ans. Radius, r = 4
Height of the cylinder, h = 14
Surface Area of cylinder = 2πr(r+h)
= 2π × 4 × (4+ 14)
= 2π × 4 × 18
= 2 × 3.14 ×4 × 18
= 628
Now, the surface area of the cylinder is 452.16 square inches.
Q2. Ram has a cylinder with a surface area of 1628π square units. Calculate the height of the cylinder if the radius of the base of the circle is 28 units.
Ans. The surface area of the cylinder, A = 1628π; radius (r) = 28; h = ?
Let us put the given values in the formula to find the height of the cylinder.
A = 2πr(r + h)
1628π = 2π × 28 × (28 + h)
1628/56 = (28 + h)
29 = (28 + h)
h = 1
Frequently Asked Question (FAQs)
Q1. What type of surface is a cylinder?
Ans. A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface or cylinder.
Q2. What is the outer surface area of the cylinder?
Ans. To calculate the lateral surface area of a cylinder with radius r and height h, follow the given steps: Calculate the perimeter of the circular base using C = 2πr. Multiply this value by the cylinder's height to get its lateral surface area, Aₗ = 2πrh.
Q3. What is the formula of the cylinder?
Ans. Cylinders volume is given by the formula, πr2h, where r is the radius of the circular base and the height of the cylinder is h.
