This sculpture designed by Mr. Newman’s colleague, Tia, is shaped like a hollow triangular pyramid. All 3 of the lateral faces are congruent and made of glass. How much glass was used to make the sculpture, if each face has a base length of 4 m and a slant height of 6 m? What is the area of each of the lateral faces? m² What is the total lateral area of the sculpture? m²

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  

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