Find the lateral area of a right triangular prism with an equilateralbase length of 5cm and height of 10cm.
Area = 3 equal rectangles + 2 equal equilateral triangles
the rectangels are easy, each is 5 by 10 or 50 cm^2
For the triangle, if you know two sides and the contained angle Ø between those sides, then the area
= (1/2)(side1)(side2)sinØ
we know it is equilateral so the angle is 60°
area of base triangle = (1/2)(5)(5)sin60°
= 25/2 (√3/2) = 25√3/4 cm^2
or
height of triangle:
h^2 + 25/4 = 25
h^2 = 75/4
h = √(75/4) = 5√3/2
area = (1/2)base x height = (1/2)(5)(5√3/2) = 25√3/4
add up the rectangles and triangles.
To find the lateral area of a right triangular prism, you need to calculate the sum of the areas of all the faces except the bases.
First, let's find the area of the triangular face. Since the base of the triangular prism is an equilateral triangle, all sides are equal. Given that the length of one side of the equilateral triangle is 5 cm, we can find the height of the triangular face using the Pythagorean theorem.
The height of the triangular face (h) can be calculated as:
h = sqrt(5^2 - (5/2)^2)
h = sqrt(25 - 6.25)
h = sqrt(18.75)
h ≈ 4.33 cm
The area of the triangular face (A_triangle) can be calculated as:
A_triangle = (base * height) / 2
A_triangle = (5 * 4.33) / 2
A_triangle = 21.65 cm²
To find the lateral area, we multiply the area of the triangular face by the number of faces, which is 2 in this case (since there are two identical triangular faces).
Lateral area = 2 * A_triangle
Lateral area = 2 * 21.65
Lateral area = 43.3 cm²
Therefore, the lateral area of the right triangular prism is 43.3 cm².