Lateral Area of a Cone
The lateral area of a cone is defined as the area covered by the curved surface of the cone. It is also called lateral surface area (LSA) or curved surface area (CSA) of a cone. A cone is a 3-D object which tapers smoothly from the flat circular base to a point called the apex. In other words, it is a shape formed by a set of line segments coming from the base that connects to a common point (apex). These line segments start from the points in the base and end at the apex.
| 1. | What is Lateral Area of a Cone? |
| 2. | Formula of Lateral Area of a Cone |
| 3. | How to Find the Lateral Area of a Cone? |
| 4. | FAQs on the Lateral Area of Cone |
What is Lateral Area of a Cone?
The lateral surface area of a cone is the amount of region occupied by the curved surface area of the cone. As a cone is a three-dimensional shape, thus the lateral surface area of the cone also lies in the three-dimensional plane. When many triangles are stacked and rotated around an axis, we get this shape known as a cone. As it has a flat base, thus it has a total surface area as well as a curved surface area. The lateral area of a cone is represented in square units, e.g., cm2, m2, in2, etc.
Formula of Lateral Area of a Cone
The formula of the lateral area of a cone is πrL, where r is the base radius and L, is the slant height. Thus, if the slant height and base radius of a cone is known, its lateral surface area (or curved surface area) can be found. We can also write the curved area of the cone in terms of the height of the cone as we know the relation between the height and slant height of a cone using the Pythagoras theorem. The relation between the height and slant height of the cone given as L = √(h2 + r2) where h is the height of the cone. Thus, lateral area of cone = πrL = πr√(h2 + r2)
How To Find the Lateral Area of a Cone?
As we learned in the previous section, the lateral area of a cone is πrL. Thus, we follow the steps shown below to find the lateral area of a cone:
- Step 1: Identify the base radius of the cone and name it r.
- Step 2: Identify its height and name it h.
- Step 3: Find the lateral surface area of a cone using the formula πrL.
- Step 4: Represent the final answer in square units.
Example: What is the lateral area of a cone having base radius = 4 units and slant height = 7 units?
Solution:Given r = 4 units and l = 7 units
As we know, lateral area of the cone = πrL
⇒ Lateral area of cone = (22/7) × 4 × 7 = 88 units2
Answer: The lateral area of the cone is 88 units2.
FAQs on the Lateral Area of a Cone
What is the Lateral Area of a Cone?
The lateral area of a cone is defined as the area that is covered by the curved surface of the cone. It is also commonly called lateral surface area (LSA) or curved surface area (CSA) of a cone. The unit of the lateral area of a cone is given in square units, e.g., cm2, m2, in2, etc.
What is the Formula of Lateral Area of a Cone?
The formula of the lateral area of a cone is given as, lateral area of a cone = πrL where "r" and "L" are radius of the cone and the slant height of the cone. Thus, it is possible to determine the value of the lateral area of a cone if we have values of both the dimensions of the cone.
What is the Formula of Lateral Area of a Cone in Terms of Height of Cone?
We know, the formula of the lateral area of a cone is given as, lateral area of a cone = πrL. We also know the relation between slant height and the height of the cone is L = √(h2 + r2). Thus, the lateral area of the cone in terms of the height of the cone is given as πr√(h2 + r2).
How to Find the Lateral Area of a Cone?
The lateral area of a cone can be determined using the following steps:
- Step 1: Identify the base radius and height of the cone.
- Step 2: Determine the lateral area of a cone using the formula πrL.
- Step 3: Once, the value of the lateral area of the cone is obtained, represent the final answer in square units.
How to Find the Radius of a Cone If the Lateral Area of a Cone is Given?
We can find the radius of a cone if the lateral area of a cone the following steps:
- Step 1: Identify the given dimensions of the cone and let the radius of the cone is "r"
- Step 2: Substitute the values in the formula πrL and obtain an equation.
- Step 3: Solve the equation for "r".
- Step 4: Once, the value of the radius of the cone is obtained, represent the final answer in units.
What Happens to the Lateral Area of a Cone When the Radius of Cone is Doubled?
The lateral area of the cone doubles when the radius of the cone is doubled as we substitute "2r" instead of "r" in the formula of the lateral area of the cone. Thus, the lateral area of the cone = πrL = π(2r)L = 2(πrL) which is two times the original value of the lateral area of the cone.
What Happens to the Lateral Area of a Cone When the Slant Height of Cone is Halved?
The lateral area of the cone halves when the radius of the cone is halved as we substitute "(L/2)" instead of "L" in the formula of the lateral area of the cone. Thus, the lateral area of the cone = πrL = πr(L/2) = (1/2) × (πrL) which is half the original value of the lateral area of the cone.