A conical tent of height 10 m and base diameter 48 m was erected by a company in a park. Find the curved surface area of the tent (In m2).
624 π
Understanding the Conical Tent Problem
The problem asks us to find the curved surface area of a conical tent given its height and base diameter. A conical tent is essentially the shape of a cone. The curved surface area of a cone is the area of the slanting side, excluding the base.
Given Parameters of the Conical Tent
- Height (h) of the tent = 10 m
- Base Diameter (d) of the tent = 48 m
Calculating the Base Radius
The base diameter is given as 48 m. The radius (r) of the base is half of the diameter.
Radius (r) = Diameter / 2
Radius (r) = $\frac{48}{2} \text{ m}$
Radius (r) = 24 m
Finding the Slant Height of the Cone
To calculate the curved surface area of a cone, we need the slant height (l). The height (h), radius (r), and slant height (l) of a cone form a right-angled triangle, where the slant height is the hypotenuse. We can use the Pythagorean theorem to find the slant height:
$\text{l}^2 = \text{r}^2 + \text{h}^2$
Substitute the values of r and h:
$\text{l}^2 = (24)^2 + (10)^2$
$\text{l}^2 = 576 + 100$
$\text{l}^2 = 676$
Now, take the square root of both sides to find l:
$\text{l} = \sqrt{676}$
$\text{l} = 26 \text{ m}$
The slant height of the conical tent is 26 m.
Calculating the Curved Surface Area of the Conical Tent
The formula for the curved surface area (CSA) of a cone is:
$\text{CSA} = \pi \cdot \text{r} \cdot \text{l}$
Substitute the calculated values of r (24 m) and l (26 m) into the formula:
$\text{CSA} = \pi \cdot 24 \cdot 26$
$\text{CSA} = 624 \pi \text{ m}^2$
Final Answer
The curved surface area of the conical tent is $624 \pi$ square meters.
Revision Table: Cone Formulas
Here's a quick summary of key formulas related to cones:
| Measurement | Formula | Variables |
|---|---|---|
| Radius (r) | d / 2 | d = diameter |
| Slant Height (l) | $\sqrt{\text{r}^2 + \text{h}^2}$ | r = radius, h = height |
| Curved Surface Area (CSA) | $\pi \text{rl}$ | r = radius, l = slant height |
| Base Area | $\pi \text{r}^2$ | r = radius |
| Total Surface Area (TSA) | $\text{CSA} + \text{Base Area} = \pi \text{rl} + \pi \text{r}^2 = \pi \text{r(l + r)}$ | r = radius, l = slant height |
| Volume (V) | $\frac{1}{3} \pi \text{r}^2 \text{h}$ | r = radius, h = height |
Additional Information: Understanding Cone Geometry
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex. The height is the perpendicular distance from the apex to the center of the base. The radius is the radius of the circular base. The slant height is the distance from any point on the circumference of the base to the apex, measured along the surface of the cone. These three measures (height, radius, slant height) are related by the Pythagorean theorem when considering the right triangle formed by the height, radius, and slant height.
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