A glass prism on a chandelier is 75mm long. A base of the prism is an equilateral triangle with side lengths of 9mm and a height of about 7.8 mm. What is approximate surface area of prism?
To find the surface area of the prism, we need to calculate the area of the three sides and the area of the two triangular ends.
1. Area of the three sides:
The three sides of the prism are identical, so we only need to calculate the area of one side and multiply it by three.
The side is in the shape of an equilateral triangle with side length 9mm and height 7.8mm.
The area of an equilateral triangle is given by the formula: A = (sqrt(3)/4) * side^2, where A is the area and side is the side length.
Plugging in the values, we have: A = (sqrt(3)/4) * 9^2
A = (sqrt(3)/4) * 81
A ≈ 70.15 mm^2
Now, we multiply by three to account for all three sides:
Total area of the three sides = 3 * 70.15 = 210.45 mm^2
2. Area of the two triangular ends:
The two ends of the prism are congruent equilateral triangles with side length 9mm.
The area of each triangular end is given by the same formula as above: A = (sqrt(3)/4) * side^2
Plugging in the values, we have: A = (sqrt(3)/4) * 9^2
A = (sqrt(3)/4) * 81
A ≈ 70.15 mm^2
Since there are two ends, we multiply this area by two.
Total area of the two triangular ends = 2 * 70.15 = 140.3 mm^2
3. Total surface area of the prism:
To find the total surface area, we add the area of the three sides and the area of the two triangular ends.
Total surface area = Total area of the three sides + Total area of the two triangular ends
Total surface area ≈ 210.45 + 140.3
Total surface area ≈ 350.75 mm^2
Therefore, the approximate surface area of the prism is approximately 350.75 mm^2.
To find the surface area of the prism, we need to calculate the surface area of each face and then sum them up.
The prism has two congruent triangular faces and three rectangular faces.
1. Let's start by calculating the surface area of one triangular face:
The base of the triangular face is an equilateral triangle with side length 9mm.
The height of the triangular face can be found using the Pythagorean theorem:
h = sqrt((side length)^2 - (base/2)^2)
= sqrt(9^2 - (9/2)^2)
= sqrt(81 - 20.25)
= sqrt(60.75)
≈ 7.8mm
The area of an equilateral triangle can be calculated using the formula:
A = (sqrt(3) / 4) * (side length)^2
= (sqrt(3) / 4) * 9^2
= (sqrt(3) / 4) * 81
= (sqrt(3) * 81) / 4
≈ 110.865mm^2
2. Next, let's calculate the surface area of one rectangular face:
One side of the rectangle is the base of the triangular face, which is 9mm.
The other side of the rectangle is the height of the prism, which is 75mm.
The area of a rectangle can be calculated using the formula:
A = (length) * (width)
= 9mm * 75mm
= 675mm^2
3. Finally, we calculate the surface area of the prism:
The total surface area is the sum of the areas of both triangular faces and three rectangular faces:
Total Surface Area = 2 * (Area of triangular face) + 3 * (Area of rectangular face)
= 2 * 110.865mm^2 + 3 * 675mm^2
= 221.73mm^2 + 2025mm^2
≈ 2246.73mm^2
Therefore, the approximate surface area of the prism is approximately 2246.73 square millimeters.