To find the surface area of a cone given its slant height and height, follow these steps:
### Step 1: Identify the given measurements
- Slant height (\(\(L\)\)) = 13 cm
- Height (\(\(H\)\)) = 12 cm
### Step 2: Calculate the radius of the base
We will use the Pythagorean theorem to find the radius (\(\(R\)\)). The relationship between the slant height, radius, and height is given by:
\(\[ L^2 = R^2 + H^2 \]\)
Rearranging this to solve for \(\(R\)\), we get:
\(\[ R = \sqrt{L^2 - H^2} \]\)
Substitute the known values:
\(\[ R = \sqrt{13^2 - 12^2} \]\)
\(\[ R = \sqrt{169 - 144} \]\)
\(\[ R = \sqrt{25} \]\)
\(\[ R = 5 \, \text{cm} \]\)
### Step 3: Calculate the surface area of the cone
The surface area (\(\(A\)\)) of a cone is given by:
\(\[ A = \pi R (R + L) \]\)
Substitute the known values:
\(\[ A = \pi \times 5 \times (5 + 13) \]\)
\(\[ A = \pi \times 5 \times 18 \]\)
\(\[ A = \pi \times 90 \]\)
### Step 4: Compute the surface area
Using the value of \(\(\pi \approx 3.14159\)\),
\(\[ A = 3.14159 \times 90 \]\)
\(\[ A \approx 282.743 \, \text{cm}^2 \]\)
### Step 5: Round the surface area to the nearest square centimetre
\(\[ A \approx 283 \, \text{cm}^2 \]\)
### Summary
- Radius of the base: 5 cm
- Exact surface area: 282.743 cm²
- Rounded surface area: 283 cm²
So, the surface area of the cone, rounded to the nearest square centimetre, is 283 cm².
