If the slant height of a cone is 60 cm and the radius of its base is 21 cm, then find its curved surface area. (use π = \({22 \ {} \over 7}\))
3960 cm2
Calculating the Curved Surface Area of a Cone
Let's find the curved surface area of a cone using the given dimensions. We are provided with the slant height and the radius of the base.
The formula for the curved surface area (CSA) of a cone is:
CSA = πrl
Where:
ris the radius of the baselis the slant height of the coneπis the mathematical constant pi
In this problem, we are given:
- Slant height,
l = 60cm - Radius of the base,
r = 21cm - We need to use
π = ${22 \over 7}$
Now, let's substitute these values into the curved surface area formula:
CSA = ${22 \over 7} \times 21 \times 60$
We can simplify the calculation:
CSA = ${22 \times (21 \over 7) \times 60}$
CSA = ${22 \times 3 \times 60}$
First, calculate $22 \times 3$:
$22 \times 3 = 66$
Now, multiply the result by 60:
CSA = ${66 \times 60}$
CSA = $3960$
So, the curved surface area of the cone is 3960 cm2.
Cone Curved Surface Area Calculation Steps
Here are the steps we followed to calculate the curved surface area:
- Identify the given values for the radius (r) and slant height (l).
- Recall or find the formula for the curved surface area of a cone: CSA = πrl.
- Substitute the given values of r, l, and the specified value of π into the formula.
- Perform the necessary multiplication and division to calculate the CSA.
- State the final answer with the correct units (cm2).
Revision Table: Cone Formulas
Here is a quick summary of important formulas related to cones:
| Formula | Description | Variables |
|---|---|---|
| Curved Surface Area (CSA) | Area of the slanted surface | πrl |
| Total Surface Area (TSA) | Area of curved surface + Area of base | πr(r + l) |
| Volume (V) | Space occupied by the cone | ${1 \over 3}\pi r^2h$ |
| Slant Height (l) | Relationship between height, radius, and slant height (Pythagorean theorem) | $\sqrt{r^2 + h^2}$ |
Additional Information: Understanding Cone Dimensions
When working with cone problems, it's helpful to understand the different dimensions:
- Radius (r): The distance from the center of the circular base to any point on its edge.
- Height (h): The perpendicular distance from the apex (tip) of the cone to the center of the base.
- Slant Height (l): The distance from the apex of the cone to any point on the circumference of the base. This is the length along the "slope" of the cone.
The height, radius, and slant height form a right-angled triangle, where the slant height is the hypotenuse. This relationship allows us to find one dimension if the other two are known, often using the Pythagorean theorem ($l^2 = r^2 + h^2$). In this specific problem, we only needed the radius and slant height for the curved surface area calculation.
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