The base of a regular pyramid is a hexagon whose sides have length 5 ft. If a lateral edge of this pyramid has length 9 ft, what is the surface area of the pyramid?
The distance from the apex to the middle of a side of the base is
sqrt(9^2 - 2.5^2) = 8.6458 ft
That equals the height h of one of the six triangular sides.
The area of one triangular side is
(1/2)*b*h = 21.615 ft^2
Multiply that by 6 for the total side area: 129.7 ft^2
You did not say if they wanted the area of the base included. That is easy enough to do.
You should be able to do the side triangles the same way I showed you in your other post.
The problem is the base.
It is a hexagon.
It is made up of 6 equilateral triangles.
For each triangle all sides are equal and all angles are 60°
Work on only one triangle
You can find the height by Pythagoras or simple trig
Then find the area of ONE triangle, then multiply by 6
To find the surface area of the pyramid, we need to calculate the area of the base and the area of the lateral faces.
First, let's calculate the area of the base. Since the base is a regular hexagon, we can use the formula for the area of a regular hexagon:
Area = (3√3/2) * s^2
where s is the length of a side of the hexagon.
Given that the sides of the hexagon have a length of 5 ft, we can substitute this value into the formula:
Area of Base = (3√3/2) * (5^2)
Next, let's calculate the area of each lateral face. Each lateral face of the pyramid is a triangle, and since the pyramid is regular, all the lateral faces have the same area.
To find the area of a triangular lateral face, we can use the formula for the area of a triangle:
Area of Triangle = (b * h) / 2
In this case, the base of the triangle is the length of the lateral edge, which is 9 ft. To find the height of the triangle, we can use the Pythagorean theorem. The height is the height of the pyramid, which is also the slant height of the triangular lateral face. We can calculate the height using the equation:
height = √ (lateral edge^2 - (apothem)^2)
Since the pyramid is regular, the apothem is the distance from the center of the base to the midpoint of a side. For a regular hexagon, the apothem is equal to the side length divided by 2√3.
apothem = 5 / (2√3)
Now we can substitute the values into the equation to find the height:
height = √ (9^2 - (5 / (2√3))^2)
Finally, we can substitute the height and base into the area formula:
Area of Triangle = (9 * height) / 2
Since there are 6 lateral faces, the total area of the lateral faces is:
Total Area of Lateral Faces = 6 * Area of Triangle
To find the surface area of the pyramid, we add the area of the base and the total area of the lateral faces:
Surface Area = Area of Base + Total Area of Lateral Faces
Now, you can plug in the values and solve the equation to find the surface area of the pyramid.