A metalworker is creating decorative points in the form of triangular pyramids to go on top of a fence she is building. She wants to know the surface area of each pyramid. Use the surface area formula to find the surface area of each pyramid based on the figure. 4.5cm 3.9cm 3.5cm. Is the answer 32.4
To calculate the surface area of a triangular pyramid, you need to find the area of the base and the area of each triangular face.
The base of the pyramid is a triangle with side lengths of 4.5 cm, 3.9 cm, and 3.5 cm. To find the area of this triangle, we can use Heron's formula or split it into two right-angled triangles.
Assuming the triangle is scalene, we'll use Heron's formula:
a = 4.5 cm
b = 3.9 cm
c = 3.5 cm
s = (a + b + c) / 2
s = (4.5 cm + 3.9 cm + 3.5 cm) / 2
s = 11.9 cm / 2
s ≈ 5.95 cm
Area of Base = √(s * (s - a) * (s - b) * (s - c))
Area of Base = √(5.95 cm * (5.95 cm - 4.5 cm) * (5.95 cm - 3.9 cm) * (5.95 cm - 3.5 cm))
Area of Base ≈ √(5.95 cm * 1.45 cm * 2.05 cm * 2.45 cm)
Area of Base ≈ √(14.3700875 cm^4)
Area of Base ≈ 3.789950766 cm^2
Now, let's find the area of each triangular face. We can find the height of the pyramid by using the Pythagorean theorem with one of the base's right-angled triangles.
Given the base triangle with side lengths of 3.9 cm, 4.5 cm, and 3.5 cm, let's use the side lengths 3.5 cm and 3.9 cm as the legs of a right-angled triangle. The hypotenuse (h) will be the height of the pyramid.
h^2 = (3.9 cm)^2 - (3.5 cm)^2
h^2 = 15.21 cm^2 - 12.25 cm^2
h^2 ≈ 2.96 cm^2
h ≈ √2.96 cm^2
h ≈ 1.723 cm
Now, we can find the area of each triangular face:
Area of Triangle = (1/2) * base * height
Area of Triangle ≈ (1/2) * 3.9 cm * 1.723 cm
Area of Triangle ≈ 3.368985 cm^2
There are four identical triangular faces on a pyramid, so the total surface area would be:
Surface Area = 4 * Area of Triangle + Area of Base
Surface Area ≈ 4 * 3.368985 cm^2 + 3.789950766 cm^2
Surface Area ≈ 13.47594 cm^2 + 3.789950766 cm^2
Surface Area ≈ 17.265890766 cm^2
Therefore, the surface area of each pyramid is approximately 17.2659 cm^2, not 32.4 cm^2.