The length of a solid prism is 10cm. It's cross section is an equilateral triangle of sides 6cm. Find the total surface area of the prism
the area is two triangles plus three rectangles, so
2(√3/4 * 6^2) + 3(10*6) = 180 + 18√3
To find the total surface area of the prism, we need to find the areas of all its faces and then add them up.
Let's start by finding the area of the triangular prism face. The cross-section is an equilateral triangle with side length 6 cm.
1. To find the area of an equilateral triangle, we can use the formula A = (√3/4) * a^2, where "a" is the side length of the triangle.
Plugging in the values, we have:
A = (√3/4) * 6^2
A = (√3/4) * 36
A = (√3/4) * 36
A = (1.732/4) * 36
A = 0.433 * 36
A = 15.588 cm^2
So, the area of one triangular face is 15.588 cm^2.
2. Now, let's find the area of the rectangular face. The dimensions of the rectangle are the length of the prism (10 cm) and the height of the equilateral triangle (6 cm).
The formula to find the area of a rectangle is A = length * width.
A = 10 cm * 6 cm
A = 60 cm^2
So, the area of one rectangular face is 60 cm^2.
3. The prism has two triangular faces and three rectangular faces.
The total surface area of the prism is the sum of these areas:
Total surface area = 2 * area of triangular face + 3 * area of rectangular face
Total surface area = 2 * 15.588 cm^2 + 3 * 60 cm^2
Total surface area = 31.176 cm^2 + 180 cm^2
Total surface area = 211.176 cm^2
Therefore, the total surface area of the prism is 211.176 cm^2.
To find the total surface area of a prism, we need to calculate the areas of all its faces, including the bases and the lateral faces.
In this case, the solid prism has an equilateral triangle as its cross section. Since the sides of the equilateral triangle are given as 6 cm, we can use the formula for the area of an equilateral triangle:
Area = sqrt(3) * (side)^2 / 4
Plugging in the values, we have:
Area = sqrt(3) * (6 cm)^2 / 4
Area = sqrt(3) * 36 cm^2 / 4
Area = sqrt(3) * 9 cm^2
Area = 9 sqrt(3) cm^2
The solid prism has two congruent triangular bases, so the total area of the bases is:
2 * Area = 2 * (9 sqrt(3) cm^2)
Total base area = 18 sqrt(3) cm^2
The solid prism also has three congruent rectangular lateral faces. The dimensions of these rectangles can be calculated as follows:
Length = Length of prism = 10 cm
Width = Side of the equilateral triangle = 6 cm
Height = Length of the rectangular face = 10 cm
The area of one rectangular face is:
Area = Length * Height = 10 cm * 10 cm = 100 cm^2
Since the solid prism has three identical rectangular faces, the total area of the lateral faces is:
3 * Area = 3 * 100 cm^2 = 300 cm^2
Finally, to find the total surface area, we add the areas of the bases and the lateral faces:
Total surface area = Total base area + Total lateral area
Total surface area = 18 sqrt(3) cm^2 + 300 cm^2
Hence, the total surface area of the solid prism is 18 sqrt(3) cm^2 + 300 cm^2, which can be simplified depending on the required level of precision.