Find the total surface area of a solid right pyramid of its height is 4 cm, and its square base is of side 6 cm.

Options

A. a) 86 sq. cm
B. b) 90 sq. cm.
C. c) 80 sq. cm.
D. d) 96 sq. cm.

Answer

To find the total surface area of a solid right pyramid with a height of 4 cm and a square base of side 6 cm, we can follow these steps:

### Step 1: Identify the dimensions of the pyramid
- Height (h) = 4 cm
- Side of the base (a) = 6 cm

### Step 2: Calculate the area of the base
The base of the pyramid is a square, so the area (A_base) can be calculated using the formula:
\[ A_{\text{base}} = a^2 \]
Substituting the value of \( a \):
\[ A_{\text{base}} = 6^2 = 36 \, \text{cm}^2 \]

### Step 3: Calculate the slant height (l)
The slant height can be found using the Pythagorean theorem. The slant height is the hypotenuse of a right triangle where one leg is the height of the pyramid and the other leg is half the length of the base.
- Half of the base side = \( \frac{a}{2} = \frac{6}{2} = 3 \, \text{cm} \)

Using the Pythagorean theorem:
\[ l = \sqrt{h^2 + \left(\frac{a}{2}\right)^2} \]
Substituting the values:
\[ l = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \, \text{cm} \]

### Step 4: Calculate the lateral surface area
The lateral surface area (A_lateral) of a pyramid can be calculated using the formula:
\[ A_{\text{lateral}} = \frac{1}{2} \times \text{perimeter of base} \times \text{slant height} \]
The perimeter of the square base is:
\[ \text{Perimeter} = 4 \times a = 4 \times 6 = 24 \, \text{cm} \]
Now substituting the values:
\[ A_{\text{lateral}} = \frac{1}{2} \times 24 \times 5 = 60 \, \text{cm}^2 \]

### Step 5: Calculate the total surface area
The total surface area (A_total) of the pyramid is the sum of the lateral surface area and the area of the base:
\[ A_{\text{total}} = A_{\text{lateral}} + A_{\text{base}} \]
Substituting the values:
\[ A_{\text{total}} = 60 + 36 = 96 \, \text{cm}^2 \]

### Final Answer
The total surface area of the solid right pyramid is \( 96 \, \text{cm}^2 \).

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