A right pyramid 15 cm high stands on a square base of side 16 cm. Find its total surface area.

Options

A. a) 376 sq. cm
B. b) 1280 sq. cm
C. c) 736 sq. cm
D. d) 800 sq. cm

Answer

To find the total surface area (TSA) of a right pyramid with a square base, we will follow these steps:

### Step 1: Identify the dimensions of the pyramid
- Height (h) = 15 cm
- Side of the square base (a) = 16 cm

### Step 2: Calculate the slant height (l) of the pyramid
The slant height can be calculated using the Pythagorean theorem. The formula for slant height in a right pyramid is:
\[ l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} \]

Substituting the values:
\[ l = \sqrt{15^2 + \left(\frac{16}{2}\right)^2} \]
\[ l = \sqrt{15^2 + 8^2} \]
\[ l = \sqrt{225 + 64} \]
\[ l = \sqrt{289} \]
\[ l = 17 \, \text{cm} \]

### Step 3: Calculate the area of one slant face
The area of one triangular slant face is given by the formula:
\[ \text{Area of one slant face} = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, the base is the side of the square base (16 cm) and the height is the slant height (17 cm):
\[ \text{Area of one slant face} = \frac{1}{2} \times 16 \times 17 \]
\[ = 8 \times 17 \]
\[ = 136 \, \text{cm}^2 \]

### Step 4: Calculate the total area of the slant faces
Since there are 4 triangular slant faces in a square pyramid:
\[ \text{Area of 4 slant faces} = 4 \times 136 \]
\[ = 544 \, \text{cm}^2 \]

### Step 5: Calculate the area of the base
The area of the square base is given by:
\[ \text{Area of base} = a^2 \]
\[ = 16^2 \]
\[ = 256 \, \text{cm}^2 \]

### Step 6: Calculate the total surface area (TSA)
The total surface area of the pyramid is the sum of the area of the base and the area of the slant faces:
\[ \text{TSA} = \text{Area of base} + \text{Area of 4 slant faces} \]
\[ = 256 + 544 \]
\[ = 800 \, \text{cm}^2 \]

### Final Answer
The total surface area of the pyramid is **800 cm²**.

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