Math Problem Statement
Solution
The image shows a triangular prism with the following dimensions:
- Height (vertical side of the triangle): 6.1 cm
- Base length of the triangle: 7.0 cm
- Side length of the triangle: 5.0 cm
To calculate the surface area of the prism, we need to find the area of its two triangular bases and the area of its rectangular lateral faces. The surface area can be calculated using the following formula:
1. Area of the Triangle:
We can use Heron's formula to calculate the area of the triangle, where the semi-perimeter is:
Now, apply Heron's formula for the area of the triangle:
2. Perimeter of the Triangle:
The perimeter of the triangle is:
3. Surface Area of Prism:
Now, let's plug everything into the surface area formula:
Let me calculate the exact surface area for you. math
Given values
a = 7.0 # base of triangle (cm) b = 5.0 # side of triangle (cm) c = 6.1 # height of triangle (cm)
Semi-perimeter of the triangle
s = (a + b + c) / 2
Area of the triangle using Heron's formula
area_triangle = math.sqrt(s * (s - a) * (s - b) * (s - c))
Perimeter of the triangle
perimeter_triangle = a + b + c
Length of the prism (height of the prism)
length_prism = 6.1
Surface area of the prism
surface_area = 2 * area_triangle + perimeter_triangle * length_prism surface_areaThe surface area of Prism A is approximately 140.19 cm².
Would you like me to assist further with other prisms or additional details? Here are some related questions:
- How do you calculate the surface area for other types of prisms?
- What is the difference between surface area and volume of a prism?
- How do you apply Heron’s formula in different scenarios?
- How can you calculate the volume of this triangular prism?
- What is the effect of changing the side lengths on the surface area?
Tip: When using Heron's formula, always ensure that the semi-perimeter is calculated correctly before applying it to find the area.
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Math Problem Analysis
Mathematical Concepts
Surface Area
Geometry
Heron's Formula
Formulas
Surface Area = 2 × Area of Triangle + Perimeter of Triangle × Length of Prism
Heron's Formula: Area of Triangle = √(s(s-a)(s-b)(s-c)), where s = (a + b + c) / 2
Theorems
Heron's Formula for the area of a triangle
Suitable Grade Level
Grades 7-9
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