Question
PLEASE answer correctly and use the details i give you to answer the question.
Two front faces of a triangular pyramid are shown. Each face shows three levels of triangle shapes: 1 at the top, 3 in the middle row, and 5 along the bottom row. The orientation of the triangles alternates from upright to inverted on the bottom two rows. The sides of the triangular pyramid are equilateral and measure 4 inches. The vertical height measures 3.646 inches.
What is the surface area of Sally’s Pyraminx? Write your answer in decimal form. Round to the nearest thousandth.
(1 point)
Two front faces of a triangular pyramid are shown. Each face shows three levels of triangle shapes: 1 at the top, 3 in the middle row, and 5 along the bottom row. The orientation of the triangles alternates from upright to inverted on the bottom two rows. The sides of the triangular pyramid are equilateral and measure 4 inches. The vertical height measures 3.646 inches.
What is the surface area of Sally’s Pyraminx? Write your answer in decimal form. Round to the nearest thousandth.
(1 point)
Answers
GPT 3.5
The surface area of Sally's Pyraminx can be calculated using the formula for the surface area of a triangular pyramid, which is given by:
Surface Area = base area + (1/2) * perimeter of base * slant height
First, we need to calculate the base area and perimeter of the base of the equilateral triangle:
Base Area = (sqrt(3) / 4) * side^2
Base Area = (sqrt(3) / 4) * 4^2
Base Area = (sqrt(3) / 4) * 16
Base Area = 4 * sqrt(3)
Perimeter of the Base = 3 * side
Perimeter of the Base = 3 * 4
Perimeter of the Base = 12
Next, we calculate the slant height using the Pythagorean theorem:
Slant Height = sqrt(height^2 + (side/2)^2)
Slant Height = sqrt(3.646^2 + 2^2)
Slant Height = sqrt(13.291 - 4)
Slant Height = sqrt(9.291)
Slant Height = 3.047
Now, we can plug in the values into the surface area formula:
Surface Area = 4 * sqrt(3) + (1/2) * 12 * 3.047
Surface Area = 4 * sqrt(3) + 18.282
Surface Area ≈ 4 * 1.732 + 18.282
Surface Area ≈ 6.928 + 18.282
Surface Area ≈ 25.21
Therefore, the surface area of Sally's Pyraminx is approximately 25.21 square inches.
Surface Area = base area + (1/2) * perimeter of base * slant height
First, we need to calculate the base area and perimeter of the base of the equilateral triangle:
Base Area = (sqrt(3) / 4) * side^2
Base Area = (sqrt(3) / 4) * 4^2
Base Area = (sqrt(3) / 4) * 16
Base Area = 4 * sqrt(3)
Perimeter of the Base = 3 * side
Perimeter of the Base = 3 * 4
Perimeter of the Base = 12
Next, we calculate the slant height using the Pythagorean theorem:
Slant Height = sqrt(height^2 + (side/2)^2)
Slant Height = sqrt(3.646^2 + 2^2)
Slant Height = sqrt(13.291 - 4)
Slant Height = sqrt(9.291)
Slant Height = 3.047
Now, we can plug in the values into the surface area formula:
Surface Area = 4 * sqrt(3) + (1/2) * 12 * 3.047
Surface Area = 4 * sqrt(3) + 18.282
Surface Area ≈ 4 * 1.732 + 18.282
Surface Area ≈ 6.928 + 18.282
Surface Area ≈ 25.21
Therefore, the surface area of Sally's Pyraminx is approximately 25.21 square inches.