Surface area of prisms questions with answers

Questions

1. Question 1Total surface area of a cuboid

   The diagram shows a cardboard box in the shape of a cuboid. Find its total surface area.

   Show the worked solution

   1. Step 1: Six faces in three equal pairs: top and bottom, front and back, the two ends$\text{TSA} = 2(25 \times 10) + 2(25 \times 12) + 2(10 \times 12)$$= 1340$ m²
2. Question 2Open boxes

   The diagram shows an open box (it has no lid), in the shape of a cuboid. Find its total surface area (outside only).

   Show the worked solution

   1. Step 1: Five faces: the base (no lid), front and back, the two ends$\text{TSA} = 7 \times 7 + 2(7 \times 3) + 2(7 \times 3)$$= 133$ cm²
3. Question 3Triangular prisms

   The diagram shows a cheese wedge in the shape of a triangular prism. Its cross-section is a right-angled triangle. Find its total surface area.

   Show the worked solution

   1. Step 1: The two ends: area of the cross-section$\text{Triangle} = \dfrac{1}{2} \times 12 \times 9$$= 54$ m²
   2. Step 2: The faces along the length: perimeter of the cross-section × length$\text{Perimeter} = 12 + 9 + 15$$= 36$ m
   3. Step 3: Total surface area = 2 × ends + perimeter × length$\text{TSA} = 2 \times 54 + 36 \times 15$$= 648$ m²
4. Question 4Trapezium and L-shaped prisms

   The diagram shows a prism whose cross-section is a trapezium. The dashed line is perpendicular to the parallel sides. Find the total surface area of the prism.

   Show the worked solution

   1. Step 1: The two ends: area of the cross-section$\text{Trapezium} = \dfrac{1}{2} \times (10 + 35) \times 12$$= 270$ m²
   2. Step 2: The faces along the length: perimeter of the cross-section × length$\text{Perimeter} = 35 + 20 + 10 + 15$$= 80$ m
   3. Step 3: Total surface area = 2 × ends + perimeter × length$\text{TSA} = 2 \times 270 + 80 \times 20$$= 2140$ m²
5. Question 5A missing length from the surface area

   The total surface area of the closed cuboid is $238$ m². Find the value of $h$.

   Show the worked solution

   1. Step 1: Total surface area = 2(lb + lh + bh)$2(11 \times 5 + 11h + 5h) = 238$$110 + 32h = 238$
   2. Step 2: Solve$32h = 128$$h = \dfrac{128}{32}$$h = 4$
6. Question 6The cost of painting (cm² to m²)

   The diagram shows a toy box in the shape of a cuboid, with a lid. Its outside, including the lid, is painted with paint that costs $6 per m². Find the cost of the paint needed.

   Show the worked solution

   1. Step 1: Six faces in three equal pairs$\text{Area} = 2(120 \times 80) + 2(120 \times 50) + 2(80 \times 50)$$= 39\,200$ cm²
   2. Step 2: Change cm² to m²: 1 m² = 100 × 100 = 10 000 cm²$\dfrac{39200}{10\,000} = 3.92$ m²
   3. Step 3: Cost = area in m² × the price per m² (in dollars)$3.92 \times 6 = 23.52$The paint costs $23.52.