The height of a cone of diameter 10 cm and slant height 13 cm, is

Options

A. 12 cm
B. 13 cm
C. `sqrt(69)cm`
D. `sqrt(194)cm`

Answer

To find the height of a cone given its diameter and slant height, we can follow these steps:

### Step 1: Identify the given values
- Diameter (D) = 10 cm
- Slant height (L) = 13 cm

### Step 2: Calculate the radius (R)
The radius (R) is half of the diameter.
\[ R = \frac{D}{2} = \frac{10 \text{ cm}}{2} = 5 \text{ cm} \]

### Step 3: Use the Pythagorean theorem
In a right triangle formed by the height (h), radius (R), and slant height (L), we can use the Pythagorean theorem:
\[ L^2 = h^2 + R^2 \]
We need to find the height (h), so we rearrange the formula:
\[ h^2 = L^2 - R^2 \]

### Step 4: Substitute the known values
Now, substitute the values of L and R into the equation:
\[ h^2 = (13 \text{ cm})^2 - (5 \text{ cm})^2 \]
\[ h^2 = 169 \text{ cm}^2 - 25 \text{ cm}^2 \]
\[ h^2 = 144 \text{ cm}^2 \]

### Step 5: Calculate the height (h)
Now, take the square root of both sides to find h:
\[ h = \sqrt{144 \text{ cm}^2} = 12 \text{ cm} \]

### Final Answer
The height of the cone is 12 cm.

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