What are the surface areas of this two figures?
a) a triangular pyramid with base of 5 cm, an altitude of the base of 8.7 cm and a slant height = 6.9 cm
b) a square pyramid with a slant height of 5.2cm and an altitude of the triangle of 9cm.
Surface area of any pyramid is base area+ 1/2 slantheight*perimeter.
a) To get base area, we use Heron's formula
Area= sqrt(s(s-a)(s-b)(s-c))
where s is one half perimeter or
Base Area= sqrt(7.5*2.5)^3)=you do it.
Slant height= sqrt (6.9^2 + (.5*8.7)^2)
Perimeter=15
b) I don't see how the altitude can be greater than the slant height.
got it!
I assume you caught my typo...
base area=sqrt(7.5(2.5)^3)
To find the surface area of a figure, we need to determine the areas of all its individual surfaces and then sum them up.
a) For the triangular pyramid, there are four surfaces: the base and three triangular faces.
1. The area of the base can be determined using the formula for the area of a triangle, which is (1/2) * base * height. In this case, the base is a triangle with a base of 5 cm and an altitude of 8.7 cm. Therefore, the area of the base is (1/2) * 5 cm * 8.7 cm = 21.75 cm^2.
2. The three triangular faces have the same base as the base of the pyramid, but their heights (or altitudes) are given by the slant height of the pyramid, which is 6.9 cm. The area of each triangular face can be calculated using the same formula as for the base. So each face has an area of (1/2) * 5 cm * 6.9 cm = 17.25 cm^2.
To get the total surface area of the triangular pyramid, we add up the areas of all the surfaces: area of the base + sum of the areas of the three triangular faces.
Total surface area = 21.75 cm^2 + 17.25 cm^2 + 17.25 cm^2 + 17.25 cm^2 = 73.5 cm^2.
b) For the square pyramid, there are five surfaces: the base and four triangular faces.
1. The area of the base is simply the square of one of its sides. Since it is a square pyramid, the base is a square, and its side length can be found using the Pythagorean theorem. The altitude of the triangle is given as 9 cm, and the slant height of the pyramid is given as 5.2 cm. The side length of the base can be found using the equation: side^2 = slant height^2 - (altitude^2/4). Plugging in the values, we get side^2 = 5.2 cm^2 - (9 cm^2/4) = 5.2 cm^2 - 20.25 cm^2 = -15.05 cm^2, which is not possible as lengths cannot be negative. This means that the given measurements for the square pyramid are not feasible, and a valid surface area cannot be calculated.
To summarize,
a) The surface area of the triangular pyramid with the given measurements is 73.5 cm^2.
b) The surface area of the square pyramid cannot be determined with the given measurements.