a regular triangular pyramid with a slant height of 9 m has a volume equal to 50 m³. Find the lateral areas of the pyramid.
If the triangular base has area A and the pyramid has height h, the volume of the pyramid is 50 = 1/3 Ah so you have 150 = Ah and h=150/A. The area of the base is A=b^2*sqrt (3) /4 where b is the length of each side of the base.
So you have a h = 150 / (b^2*sqrt (3) /4) = 200sqrt (3) / b^2
The edge length e, slant heights, and base length b satisfy e^2 = h^2 + b^2 /3 and s^2 = h^2 +b^2 /12. e^2 -s^2 = b^2 /4
Substituting into the equation s^2 = h^2 +b^2 /12 for h^2 = 120000b^4:
9^2 =120000 / b^4 +b^2 /12. This gives b = 6.26836.
Each of the three lateral triangles has an area BS /2 so the total lateral area of the pyramid is 3bs/2 = 3 (6.26836) (9) /2 = 84.62286.
You could find h using 9^2 = h^2 +6.26836^2 /12, then he = 8.81621.
Answer is 81 square root of 3 sq.m
Regular triangular pyramid has 6 cm long base edge and slant height k=9 cm. Find the lateral area of the pyramid.
hi.... i still cannot undertand how he gets those equation about the edged lengths, slant height and side ..plsss enlightem me
To find the lateral area of a triangular pyramid, you need to calculate the sum of the areas of all the triangular faces. Here's how you can do it:
1. First, determine the base area of the pyramid. Since it's a regular triangular pyramid, the base is an equilateral triangle. Let's say the base side length is "s".
2. To find the base area, you can use the formula for the area of an equilateral triangle, which is given as:
Base Area (A) = (sqrt(3) / 4) * s^2
3. Next, you need to find the slant height (l) of the pyramid. It's given as 9 m in the question.
4. The height (h) of the pyramid can be found by applying the Pythagorean theorem to one of the triangular faces. The height forms a right triangle with half the base side (s/2), the slant height (l), and the height (h). The Pythagorean theorem states that the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse. In this case, it can be represented as:
(s/2)^2 + h^2 = l^2
5. Solve the above equation to find the height (h).
6. Once you have the height (h), you can find the area of each triangular face using the formula:
Triangular Area = (1/2) * s * h
7. Since the pyramid has four triangular faces, you need to multiply the area of one triangular face by four to get the total lateral area.
That's how you can find the lateral area of the pyramid.