Question
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L-shaped prism. The base has side lengths labeled 1 cm, 3 cm, 5 cm, 1 cm and 4 cm. The height of the prism is 6 cm.
The surface area of this prism is
cm
2
.
L-shaped prism. The base has side lengths labeled 1 cm, 3 cm, 5 cm, 1 cm and 4 cm. The height of the prism is 6 cm.
The surface area of this prism is
cm
2
.
Answers
Answered by
GPT-4o mini
To calculate the surface area of an L-shaped prism, we first need to understand the shape of the base and then incorporate the height of the prism.
1. **Calculate the area of the L-shaped base**:
- The piece can be split into two rectangles:
- Rectangle 1: Dimensions 1 cm (height) x 3 cm (width)
- Rectangle 2: Dimensions 4 cm (height) x 1 cm (width)
- The dimensions in the L-shape add up:
- For the first rectangle, Area = 1 cm * 3 cm = 3 cm²
- For the second rectangle adjacent to the first rectangle, the dimension on one side includes 1 cm and on the other side 4 cm. The area of the 2nd rectangle would use only the overlapping part to cover the full area, which results as follows:
- The larger rectangle that encloses the total "L":
- Total width (of larger) = 3 cm (from rectangle 1) + 1 cm (from rectangle 2)
- Total height = max height of 1 cm and 4 cm = 4 cm,
- Area of this larger rectangle = 4 cm * 4 cm = 16 cm²
- Overlapping area = 1 cm * 1 cm = 1 cm² to subtract
- Hence, the area of the L-shape base = 16 cm² - 1 cm² = 15 cm²
2. **Calculate the lateral surface area**:
- The lateral surface area of the prism can be calculated using the perimeter of the base times the height.
- First, calculate the perimeter of the L-shape:
- Perimeter = 1 cm (left side) + 3 cm (top side) + 1 cm (right vertical side adjacent to 1 cm) + 4 cm (additional length of rectangle 2) + 1 cm (bottom length) + 1 cm (vertical side of the base)
- Total perimeter = 1 + 3 + 4 + 1 + 1 + 1 = 11 cm
- The lateral surface area = Perimeter * Height = 11 cm * 6 cm = 66 cm²
3. **Calculate the total surface area**:
- Total surface area = Lateral surface area + 2 * Area of base
- Total surface area = 66 cm² + 2 * 15 cm² = 66 cm² + 30 cm² = 96 cm²
So, the surface area of the L-shaped prism is 96 cm².
1. **Calculate the area of the L-shaped base**:
- The piece can be split into two rectangles:
- Rectangle 1: Dimensions 1 cm (height) x 3 cm (width)
- Rectangle 2: Dimensions 4 cm (height) x 1 cm (width)
- The dimensions in the L-shape add up:
- For the first rectangle, Area = 1 cm * 3 cm = 3 cm²
- For the second rectangle adjacent to the first rectangle, the dimension on one side includes 1 cm and on the other side 4 cm. The area of the 2nd rectangle would use only the overlapping part to cover the full area, which results as follows:
- The larger rectangle that encloses the total "L":
- Total width (of larger) = 3 cm (from rectangle 1) + 1 cm (from rectangle 2)
- Total height = max height of 1 cm and 4 cm = 4 cm,
- Area of this larger rectangle = 4 cm * 4 cm = 16 cm²
- Overlapping area = 1 cm * 1 cm = 1 cm² to subtract
- Hence, the area of the L-shape base = 16 cm² - 1 cm² = 15 cm²
2. **Calculate the lateral surface area**:
- The lateral surface area of the prism can be calculated using the perimeter of the base times the height.
- First, calculate the perimeter of the L-shape:
- Perimeter = 1 cm (left side) + 3 cm (top side) + 1 cm (right vertical side adjacent to 1 cm) + 4 cm (additional length of rectangle 2) + 1 cm (bottom length) + 1 cm (vertical side of the base)
- Total perimeter = 1 + 3 + 4 + 1 + 1 + 1 = 11 cm
- The lateral surface area = Perimeter * Height = 11 cm * 6 cm = 66 cm²
3. **Calculate the total surface area**:
- Total surface area = Lateral surface area + 2 * Area of base
- Total surface area = 66 cm² + 2 * 15 cm² = 66 cm² + 30 cm² = 96 cm²
So, the surface area of the L-shaped prism is 96 cm².