Use the image to answer the question.
An illustration shows a triangular pyramid. The perpendicular height of the base measures 3.9 centimeters and its sides measure 4.5 centimeters. The slant height measures 3.5 centimeters.
Find the surface area of the solid figure with an equilateral triangle base.
(1 point)
To find the surface area of the triangular pyramid, we need to calculate the area of the three triangular faces and the area of the base.
First, we calculate the area of the equilateral triangle base:
Area = (sqrt(3) / 4) * s^2
Area = (sqrt(3) / 4) * 4.5^2
Area = (sqrt(3) / 4) * 20.25
Area = 11.691
Next, we calculate the area of one of the triangular faces:
Area = (1 / 2) * base * height
Area = (1 / 2) * 4.5 * 3.5
Area = 7.88
Now, we can calculate the total surface area of the triangular pyramid:
Total surface area = 3 * Area of triangular face + Area of base
Total surface area = 3 * 7.88 + 11.691
Total surface area = 23.64 + 11.691
Total surface area = 35.331 square centimeters
Therefore, the surface area of the triangular pyramid with an equilateral triangle base is 35.331 square centimeters.
Wrong
I apologize for the error in my previous response. Let's correct the calculation:
To find the surface area of the triangular pyramid, we need to calculate the area of the three triangular faces and the area of the base.
First, let's calculate the area of the equilateral triangle base:
Area = (sqrt(3) / 4) * side^2
Area = (sqrt(3) / 4) * 4.5^2
Area = (sqrt(3) / 4) * 20.25
Area = 20.78 square centimeters
Next, we calculate the area of one of the triangular faces:
Using the Pythagorean theorem, we can find the height of the triangular face:
h = sqrt(3.5^2 - (4.5/2)^2)
h = sqrt(12.25 - 5.0625)
h = sqrt(7.1875)
h ≈ 2.68 cm
Area of one of the triangular faces = (1/2) * base * height
Area = (1/2) * 4.5 * 2.68
Area ≈ 6.78 square centimeters
Total surface area = 3 * Area of triangular face + Area of base
Total surface area = 3 * 6.78 + 20.78
Total surface area = 20.34 + 20.78
Total surface area = 41.12 square centimeters
Therefore, the correct surface area of the triangular pyramid with an equilateral triangle base is approximately 41.12 square centimeters.
Inco
I apologize for the confusion in my previous responses. Let's calculate the surface area of the triangular pyramid correctly this time.
Given:
Base side length (s) = 4.5 cm
Height of the base = 3.9 cm
Slant height = 3.5 cm
1. Find the area of the equilateral triangle base:
Area = (sqrt(3) / 4) * s^2
Area = (sqrt(3) / 4) * 4.5^2
Area = (sqrt(3) / 4) * 20.25
Area = 9.84 square cm
2. Find the area of one of the triangular faces using the base and slant height:
Area of one triangular face = (1/2) * Base * Slant height
Area of one triangular face = (1/2) * 4.5 * 3.5
Area of one triangular face = 7.875 square cm
3. Find the total surface area of the triangular pyramid:
Total surface area = Area of base + 3 * Area of one triangular face
Total surface area = 9.84 + 3 * 7.875
Total surface area = 9.84 + 23.625
Total surface area = 33.465 square cm
Therefore, the surface area of the triangular pyramid with an equilateral triangle base is 33.465 square centimeters. Thank you for your patience.