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Find the volume of a square pyramid with a base length of [tex]\frac{1}{2} \, \text{ft}[/tex] and a height of [tex]\frac{1}{4} \, \text{ft}[/tex].

Sagot :

To find the volume of a square pyramid, we use the formula:

[tex]\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \][/tex]

Given:
- The base length of the pyramid is [tex]\( \frac{1}{2} \, \text{ft} \)[/tex].
- The height of the pyramid is [tex]\( \frac{1}{4} \, \text{ft} \)[/tex].

### Step 1: Calculate the base area
The base of the pyramid is a square, so the area of the base (A) is given by:

[tex]\[ \text{Base Area} = \text{side}^2 \][/tex]

Here, the side length is [tex]\( \frac{1}{2} \, \text{ft} \)[/tex].

[tex]\[ \text{Base Area} = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \, \text{ft}^2 \][/tex]

### Step 2: Calculate the volume of the pyramid
Using the formula for the volume of a square pyramid:

[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]

Substitute the base area and height:

[tex]\[ V = \frac{1}{3} \times \frac{1}{4} \, \text{ft}^2 \times \frac{1}{4} \, \text{ft} \][/tex]

[tex]\[ V = \frac{1}{3} \times \frac{1}{16} \, \text{ft}^3 \][/tex]

[tex]\[ V = \frac{1}{48} \, \text{ft}^3 \][/tex]

### Step 3: Simplify the Volume Expression

The simplified volume is:

[tex]\[ V = 0.020833333333333332 \, \text{ft}^3 \][/tex]

Therefore, the volume of the square pyramid is [tex]\(\boxed{0.020833333333333332 \, \text{ft}^3}\)[/tex].
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