How to find the surface area of a square pyramid?
Understand the Problem
The question is asking for the method to calculate the surface area of a square pyramid, which involves understanding the formulas for area of both the square base and the triangular faces.
Answer
Total Surface Area
Answer for screen readers
The final answer is the Total Surface Area
Steps to Solve
- Calculate the area of the square base
The area of the square base can be found using the formula:
$$ ext{Area}_{ ext{base}} = s^2 $$
where $ s $ is the length of a side of the square base.
- Calculate the area of one triangular face
The triangular faces are isosceles triangles. The area of one triangle can be calculated using the formula:
$$ ext{Area}_{ ext{triangle}} = rac{1}{2} imes ext{base} imes ext{height} $$
The base of one triangle is $ s $, and the height can be found using the Pythagorean theorem if the slant height $ l $ is known:
$$ ext{height}_{ ext{triangle}} = rac{l^2 - rac{s^2}{4}}{s} $$
- Calculate the area of four triangular faces
After finding the area of one triangular face, multiply it by 4 to get the total area of the triangular faces:
$$ ext{Area}{ ext{triangles}} = 4 imes ext{Area}{ ext{triangle}} $$
- Add the areas together
Finally, add the area of the square base and the total area of the triangular faces to find the total surface area:
$$ ext{Surface area} = ext{Area}{ ext{base}} + ext{Area}{ ext{triangles}} $$
The final answer is the Total Surface Area
More Information
A square pyramid has one square base and four triangular sides. The surface area is the sum of the area of all these shapes.
Tips
A common mistake is to forget to include all four triangular faces when calculating their total area.