A regular triangular pyramid has an altitude of 9m and a volume of 46.8cu. meters. Find the length of the edges and the lateral area of the pyramid.

please answer this

V = 1/3 base * height

area of the base = 46.8/3 = 15.6
the base is a equilateral triangle.
the area of which is sqrt(3)/4 * edge^2
edge = sqrt(15.6*4/sqrt 3) = 6.002 m
The mid-point of the triangle is 2/3 of the way from the angle to the base.
sqrt(3)/3 * 6 = 2 sqrt 3 ~ 3.46 m
the length of the other 3 edges then using the Pythagorean theorem.
12+81 = edge^2
sqrt 93 ~ 9.64 m
the slant height...
3^2 + slant^2 = 93
slant height = sqrt 84 ~ 9.165 m

lateral area = 3/2 * 6.002 * 9.165 = 82.5 m^2

Why did the triangular pyramid go to the doctor? It was feeling a little obtuse!

Now, let's crunch some numbers. To find the volume of a triangular pyramid, we use the formula V = (1/3) * base area * height. In this case, we know the volume is 46.8 cu. meters and the height is 9 m, so we can rewrite the formula as 46.8 = (1/3) * base area * 9.

Simplifying that equation, the base area would be 46.8 * 3 / 9 = 15.6 sq. meters.

Now, let's find the length of the edges. Since it's a regular triangular pyramid, that means all its faces (including the base) are equilateral triangles. The formula to find the length of the edges is l = sqrt(2 * base area / sqrt(3)), where "l" is the length of each edge.

Plugging in the known values, we get l = sqrt(2 * 15.6 / sqrt(3)).

Calculating that, we find l ≈ 4.61 m.

Lastly, to find the lateral area of the pyramid, we need to find the area of each triangle face and sum them up. Since each face is an equilateral triangle, with the length of each edge being 4.61 m, we can use the formula A = (√3 / 4) * l^2 to find the area of each face.

So, the lateral area of the pyramid would be A = 4 * (√3 / 4) * (4.61)^2.

Calculating that, we find the lateral area ≈ 79.41 sq. meters.

I hope these calculations didn't make your head spin like a pyramid top!

To find the length of the edges of the regular triangular pyramid, we can use the formula:

Edge length = (3 * Volume / (sqrt(3) * Altitude))

Substituting the given values:

Edge length = (3 * 46.8 / (sqrt(3) * 9))

Edge length = (140.4 / (1.732 * 9))

Edge length = 8.097

Therefore, the length of each edge of the regular triangular pyramid is approximately 8.097m.

To find the lateral area of the pyramid, we can calculate the area of each triangular face and then sum them up:

Lateral area = (Number of faces * Area of each triangular face)

Since a regular triangular pyramid has 4 triangular faces, the lateral area can be calculated as:

Lateral area = 4 * Area of a triangular face

The formula to find the area of an equilateral triangle is:

Area = (sqrt(3) / 4 * (side length)^2)

Substituting the given edge length:

Area = (sqrt(3) / 4 * (8.097)^2)

Area = (1.732 / 4 * 65.57)

Area = 112.250

Therefore, the lateral area of the regular triangular pyramid is approximately 112.250 square meters.

To find the length of the edges and the lateral area of a regular triangular pyramid, we need to use the given information about the altitude and volume.

Let's start with finding the length of the edges:
1. The regular triangular pyramid has a regular triangular base. Each side of the base is the same length, which we'll call "b".
2. By visualizing the pyramid, we can draw a height, also known as the altitude, from the apex (top vertex) to the center of the base. In this case, the height is given as 9 meters.
3. Since we have a regular triangular pyramid, the height divides the base into two equal right-angled triangles.
4. The length of the altitude creates a right-angled triangle with one leg of the right angle as half of the base, which is "b/2".
5. The hypotenuse of this right-angled triangle is the length of one edge of the pyramid, which we'll call "e".
6. We can use the Pythagorean theorem to find the length of the edges:
e² = (b/2)² + 9²

Next, let's find the lateral area of the pyramid:
1. The lateral area is the sum of the areas of the triangular faces excluding the base.
2. Each triangular face has a base that is the length of one edge and a height that is the given altitude of 9 meters.
3. The area of a triangle is given by the formula: (1/2) * base * height.
4. Since the base of each triangular face is the length of one edge, we can use the length we found earlier.
5. The total lateral area is then the area of one triangular face multiplied by the number of faces in the pyramid.

Let's proceed with calculating the length of the edges and the lateral area of the pyramid using the given information.