Question

a square pyramid has a base with a side length of 3 feet and lateral with a height of 6 feet. what is the area of the pyramid. A.9 square feet B. 27 square feet C. 36 square feet D. 45 square feet​

Answer 1

D. 45 square feet

Explanation:

To find the surface area of a square pyramid, we need to find the area of the base and the area of the four triangular lateral faces.

Finding the area of the base

First, let's find the area of the base (square).

`\boxed{\begin{minipage}{7 cm}Base area = side length $*$ side length\\ \\Base area = 3 $*$ 3 \\ \\Base area = 9 square feet\end{minipage}}`

Finding the area of the triangular lateral faces

Now, let's find the area of the triangular lateral faces.

Finding the area of one triangular lateral face

As stated in the problem, the slant height of the triangle is 6 feet. We can find the area of one triangular lateral face:

`\text{Triangle area = $(1)/(2)$ $*$ base $*$ height}\\\\\text{Triangle area = $(1)/(2)$ $*$ 3 $*$ 6} \\\\\text{Triangle area = 9 square feet}`

Finding the total lateral area

Since there are four triangular lateral faces, we need to multiply this area by 4:

`\text{Total lateral area = 4 $*$ 9}\\\\\text{Total lateral area = 36 square feet}`

Total surface area

Finally, we add the base area and the total lateral area to find the total surface area of the pyramid:

`\text{Total surface area = base area + total lateral area}\\\\\text{Total surface area = 9 + 36}\\\\\text{Total surfacel area = 45 square feet}`

Conclusion

So, the area of the square pyramid is 45 square feet. Which is option D.

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Answer 2

D. 45 square feet

Explanation:

You want the surface area of a square base pyramid with a side length of 3 feet and a slant height of 6 feet.

Area formulas

The area of a triangular face is given by ...

A = 1/2bh . . . . . base b and height h

The area of the square base is given by ...

A = s²

The total surface area of the pyramid is the area of the base plus the areas of the 4 triangular faces:

total area = s² +4(1/2bh) . . . . where b = s

This can be simplified to ...

total area = s(s +2h) . . . . . area of a square pyramid with slant height h

Application

For the pyramid with a square base 3 ft on a side, and a slant height of 6 ft, the total area is ...

total area = (3 ft)(3 ft + 2·6 ft) = (3 ft)(15 ft) = 45 ft²

The area of the pyramid is 45 square feet.

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