Question
a metalworker is creating decorative points in the form of a triangular pyramid to go on top of a fence she is building. She wants to know the surface area of each pyramid. 4.5 cm, 3.9 cm, 3.5 cm. Use the surface area formula to find the surface area of each pyramid based on the figure.
___ cm^2
___ cm^2
Answers
GPT 3.5
To find the surface area of a triangular pyramid, we need to calculate the areas of the base and the triangular faces.
First, let's find the area of the base:
The base of a triangular pyramid is a triangle. We can use the formula for the area of a triangle, which is 0.5 * base * height. Since we don't know the height of the triangle, we need to calculate that first.
To find the height of the triangle, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
In this case, the lengths of the two known sides are 3.9 cm and 3.5 cm. Let's label them as a and b.
a^2 + b^2 = c^2
3.9^2 + 3.5^2 = c^2
15.21 + 12.25 = c^2
27.46 = c^2
c ≈ √27.46
c ≈ 5.24 cm
Now that we have the height (c) of the triangle, we can calculate the area of the base:
base = 4.5 cm
height = c = 5.24 cm
area = 0.5 * base * height
area = 0.5 * 4.5 cm * 5.24 cm
area ≈ 11.79 cm^2
Next, let's find the areas of the triangular faces:
The triangular faces of a pyramid are all identical, so we only need to calculate the area of one of them.
We can use the formula for the area of a triangle again:
area = 0.5 * base * height
In this case, the base of the triangle is the same as the base of the pyramid, which is 4.5 cm. The height of the triangle is the height of the pyramid, which we already found to be 5.24 cm.
area = 0.5 * 4.5 cm * 5.24 cm
area ≈ 11.79 cm^2
Since there are four triangular faces, the total surface area of each pyramid would be:
surface area = 4 * area
surface area = 4 * 11.79 cm^2
surface area ≈ 47.16 cm^2
So, the surface area of each pyramid is approximately 47.16 cm^2.
First, let's find the area of the base:
The base of a triangular pyramid is a triangle. We can use the formula for the area of a triangle, which is 0.5 * base * height. Since we don't know the height of the triangle, we need to calculate that first.
To find the height of the triangle, we can use the Pythagorean theorem:
a^2 + b^2 = c^2
In this case, the lengths of the two known sides are 3.9 cm and 3.5 cm. Let's label them as a and b.
a^2 + b^2 = c^2
3.9^2 + 3.5^2 = c^2
15.21 + 12.25 = c^2
27.46 = c^2
c ≈ √27.46
c ≈ 5.24 cm
Now that we have the height (c) of the triangle, we can calculate the area of the base:
base = 4.5 cm
height = c = 5.24 cm
area = 0.5 * base * height
area = 0.5 * 4.5 cm * 5.24 cm
area ≈ 11.79 cm^2
Next, let's find the areas of the triangular faces:
The triangular faces of a pyramid are all identical, so we only need to calculate the area of one of them.
We can use the formula for the area of a triangle again:
area = 0.5 * base * height
In this case, the base of the triangle is the same as the base of the pyramid, which is 4.5 cm. The height of the triangle is the height of the pyramid, which we already found to be 5.24 cm.
area = 0.5 * 4.5 cm * 5.24 cm
area ≈ 11.79 cm^2
Since there are four triangular faces, the total surface area of each pyramid would be:
surface area = 4 * area
surface area = 4 * 11.79 cm^2
surface area ≈ 47.16 cm^2
So, the surface area of each pyramid is approximately 47.16 cm^2.