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