Question

The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are of equal lengths. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then determine the total area, in sq cm, of all six surfaces of the pillar?

• A. 1340
• B. 1300
• C. 1550
• D. 1520
• E. 1480

Hint:

For a prism, the total surface area is $2 \times \text{Base Area} + \text{Perimeter of Base} \times \text{Height}$.

Answer 1

Answer: (E)

Step 1: Understand the Shape:
The pillar is a prism. The base is an isosceles trapezium with parallel sides $a=20$ cm, $b=10$ cm, height (distance between parallels) $h=12$ cm. The pillar height (prism height) $H=20$ cm.
Step 2: Find the Length of Non-Parallel Sides:
For an isosceles trapezium, the projection of each non-parallel side on the base is $\frac{20-10}{2} = 5$ cm. Thus, the non-parallel side length $l = \sqrt{h^2 + 5^2} = \sqrt{12^2 + 5^2} = \sqrt{144+25} = \sqrt{169} = 13$ cm.
Step 3: Calculate Surface Area:
The pillar has 6 surfaces: top, bottom, and 4 lateral faces. Area of top = area of base (trapezium) = $\frac{1}{2} \times (a+(b) \times h_{trap} = \frac{1}{2} \times (20+10) \times 12 = \frac{1}{2} \times 30 \times 12 = 180$ sq cm. Area of bottom = same = 180 sq cm.
Step 4: Lateral Surface Area:
The lateral surfaces are rectangles: sides of the prism with dimensions: (side length of trapezium) $\times$ (pillar height). Lateral area = Perimeter of base $\times$ Height of pillar. Perimeter of trapezium = $20 + 10 + 13 + 13 = 56$ cm. Lateral area = $56 \times 20 = 1120$ sq cm.
Step 5: Total Surface Area:
Total area = Top + Bottom + Lateral = $180 + 180 + 1120 = 1480$ sq cm.
Step 6: Final Answer:
The total area of all six surfaces is 1480 sq cm.