Math Problem Statement
what is the surface area of this shape
Solution
The given shape appears to be a triangular prism. To calculate the surface area of this shape, we need to sum the areas of all the faces of the prism.
The triangular prism consists of:
- Two triangular faces
- Three rectangular faces
Step 1: Calculate the area of the triangular faces
Each triangular face has a base of 7.2 cm and a height of 3.5 cm (as indicated in the diagram).
The area of a triangle is given by:
Substituting the values:
So, the area of one triangular face is , and since there are two triangular faces, the total area for both triangles is:
Step 2: Calculate the area of the rectangular faces
There are three rectangular faces, and their areas can be calculated as follows:
-
Front face (base × height): The front face is a rectangle with dimensions 8 cm × 5 cm.
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Bottom face (base × side length): The bottom face is a rectangle with dimensions 8 cm × 7.2 cm.
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Slant face (slant height × side length): The slant face is a rectangle with dimensions 8 cm × 5 cm (slant height).
Step 3: Add up the areas of all faces
Now, summing the areas of all the faces:
Thus, the total surface area of the prism is 162.8 cm².
Do you want any details or have any questions?
Here are five related questions to expand your understanding:
- How would the surface area change if the height of the triangle doubled?
- Can you calculate the volume of this triangular prism using the given dimensions?
- What formula would you use to find the surface area if the base was a different polygon?
- How does the slant height of the triangle affect the overall surface area?
- Can you explain how surface area differs from lateral surface area in a prism?
Tip: When calculating the surface area of a 3D shape, always ensure you account for every face, especially if the shape is irregular.
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Math Problem Analysis
Mathematical Concepts
Geometry
Surface Area
3D Shapes
Prisms
Formulas
Area of a triangle: A = 1/2 * base * height
Area of a rectangle: A = length * width
Surface area of a prism: Total surface area = sum of areas of all faces
Theorems
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Suitable Grade Level
Grades 7-9
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