Find the lateral area of a regular hexagonal pyramid with base edge of 10 in and lateral edge of 13 in.
you can find the height of each triangle ...
h^2 + 5^2 = 13^2
h = 12
Area of one such triangle = (1/2)base x heigh = (1/2)(10)(12) = 60
there will be 6 such triangles
You don't say if you include the base in the surface area, if so ...
then you can find the area of one of the six equilateral triangles that form the base.
1/2 PL
P=10*6 because all sides of the hexagon r 10
L=9
60*9/2
270
Well, well, well, we have ourselves a regular hexagonal pyramid here. Let's get to the bottom of it, shall we?
To find the lateral area, we need to calculate the total area of all the triangular faces that make up the sides of our pyramid.
Now, each face of a hexagonal pyramid is an equilateral triangle, meaning all three sides are equal. Lucky for us, we know the length of the lateral edge is 13 in.
So, let's calculate the height of this equilateral triangle. We know the base edge is 10 in, and we can use some Pythagorean magic to find the height. The height squared plus half the base squared equals the lateral edge squared.
H = √(13^2 - (10/2)^2)
H = √(169 - 25)
H = √144
H = 12 in
Now that we have the height, we can calculate the area of one equilateral triangle using the formula: (side length)^2 * (√3/4).
Area of one triangle = (10^2) * (√3/4)
Area of one triangle = 100 * (√3/4)
Area of one triangle = 25√3 in^2
Since we have six equilateral triangles making up the sides of our hexagonal pyramid, we can simply multiply the area of one triangle by 6.
Lateral area = 6 * 25√3 in^2
Lateral area = 150√3 in^2
So, my friend, the lateral area of this regular hexagonal pyramid is 150√3 square inches.
To find the lateral area of a regular hexagonal pyramid, you can use the formula:
Lateral Area = (perimeter of base) x (slant height) / 2
First, let's find the perimeter of the base of the pyramid. Since it is a regular hexagon, all of its sides are equal.
Perimeter of hexagon = 6 x (length of one side)
In this case, the length of one side (base edge) is given as 10 inches.
Perimeter of hexagon = 6 x 10 = 60 inches
Now, we need to find the slant height of the pyramid. The lateral edge is given as 13 inches. The slant height is the height of each of the triangular faces of the pyramid.
Using the Pythagorean theorem, we can find the slant height:
Slant height = √((lateral edge^2) - (apothem^2))
In a regular hexagon, the apothem is the distance from the center to the midpoint of one of the sides, which is also the height of an equilateral triangle with side length equal to the base edge.
The apothem of a regular hexagon = (base edge) / 2
In this case, the apothem = 10 / 2 = 5 inches
Slant height = √((13^2) - (5^2)) = √(169 - 25) = √144 = 12 inches
Now, we can calculate the lateral area using the formula mentioned earlier:
Lateral Area = (perimeter of base) x (slant height) / 2
Lateral Area = 60 inches x 12 inches /2 = 720 square inches
Therefore, the lateral area of the regular hexagonal pyramid is 720 square inches.
To find the lateral area of a regular hexagonal pyramid, you need to know the length of the lateral edge and the length of one of the base edges. Once you have this information, you can use the formula for the lateral area of a pyramid.
The lateral area of a pyramid is given by the formula:
Lateral Area = (Perimeter of base) × (Lateral edge length) / 2
For a regular hexagonal pyramid, the perimeter of the base can be found by multiplying the length of one of the base edges by 6 (as there are 6 sides in a regular hexagon). In this case, the length of one of the base edges is given as 10 inches, so the perimeter of the base is 6 × 10 = 60 inches.
Now that you have the perimeter of the base (60 inches) and the lateral edge length (13 inches), you can use these values in the formula to find the lateral area:
Lateral Area = (Perimeter of base) × (Lateral edge length) / 2
= 60 × 13 / 2
= 780 / 2
= 390 square inches
Therefore, the lateral area of the regular hexagonal pyramid is 390 square inches.