Answer: Each lateral face would have an area measuring 11.32 square metres.
(2) The total lateral area is 40.88 square metres.
Step-by-step explanation: A three-sided triangular prism as stated in the question would have triangular faces on all sides and since the one in the question is described as a hollow triangular pyramid, we can safely conclude that each surface is a flat/plane shape. Also, each of the triangular surfaces has a base length of 4 metres and a slant height of 6 metres. The surface area is given as
Area of a triangle = 1/2 * base * height
However, the height given is the slant height. Therefore to calculate the vertical height, we shall apply the Pythagoras' theorem, which means the triangular surface would have to be divided into two halves by a perpendicular line that splits the base length into two equal sides. This results in a right angled triangle with the hypotenuse measuring 6 metres and one of the two other sides measuring 2 metres. We now have
AC² = BC² + AB²
Where AC is the hypotenuse 6, BC is one of the other sides 2, and AB is the unknown (slant height).
6² = 2² + AB²
36 = 4 + AB²
Subtract 4 from both sides of the equation
32 = AB²
Add the square root sign to both sides of the equation
√32 = √AB²
5.66 = AB
The area of the triangular surface now becomes;
Area = 1/2*base*height
Area = 1/2 x (4 x 5.66)
Area = 1/2 x 22.64
Area = 11.32 metres²
The total of all three surfaces would be derived as
Surface areas = 11.32 x 3
Surface areas = 33.96 metres²
For the base area, with all three sides measuring 4 metres each, the area would be derived but first we need to calculate the slant height first.
With a line drawn perpendicular to one of the sides, we now have a right angled triangle with the hypotenuse as 4, one of the sides as 2, and the other side unknown (slant height).
AC² = AB² + BC²
4² = 2² + BC²
16 = 4 + BC²
Subtract 4 from both sides of the equation
12 = BC²
Add the square root sign to both sides of the equation
√12 = √BC²
3.46 = BC
The area of the base triangle now becomes;
Area = 1/2*base*height
Area = 1/2 x (4 x 3.46)
Area = 1/2 x (13.84)
Area = 6.92
The total lateral area would now become
Lateral area = Surface area + Base area
Lateral area = 33.96 + 6.92
Lateral area = 40.88 metres²