Math Problem Statement
Use Pythagoras’ theorem to determine any unknown side lengths and find the total surface area of these solids, correct to one decimal place.
Solution
To solve this problem, we will use the Pythagorean theorem to determine any unknown side lengths of the triangular faces, and then we'll calculate the total surface area of the solid.
Step 1: Identify the Shape and Given Dimensions
The image shows a triangular pyramid (tetrahedron) where:
- One right triangle has sides 7, 8, and 10 units.
- The other triangle also appears to be a right triangle, with a side length of 7 units and a height of 8 units.
Step 2: Calculate the Unknown Side Lengths
First Triangle (Right Triangle):
The first triangle is a right triangle with sides:
- (the hypotenuse)
This triangle is already defined, so no further calculation is needed here.
Second Triangle:
This second triangle is also a right triangle:
- One leg is 7 units
- The other leg is 8 units
We need to calculate the hypotenuse () of this triangle using the Pythagorean theorem:
Step 3: Calculate the Surface Area
The tetrahedron consists of four triangular faces:
- Two triangles with sides units.
- Two triangles with sides units.
3.1: Area of Each Triangle
-
Triangle with sides 7, 8, and 10: Since it's a right triangle, the area is:
-
Triangle with sides 7, 8, and 10.6: Similarly, since this is also a right triangle:
3.2: Total Surface Area
The tetrahedron has four triangular faces, so the total surface area is the sum of the areas of these four triangles:
Final Answer
The total surface area of the solid is 112 square units, rounded to one decimal place.
Would you like more details or have any questions? Here are some related questions:
- How do you calculate the surface area of other 3D shapes like pyramids and cylinders?
- How would you approach this problem if the triangles were not right triangles?
- What is the difference between a prism and a pyramid in terms of surface area calculation?
- How can you find the volume of this tetrahedron?
- How can the Pythagorean theorem be applied to find distances in 3D space?
- What if the given sides were in different units—how would you convert them?
- How do you calculate the area of an irregular polygon?
- What are some real-life applications of calculating surface area?
Tip: When working with 3D shapes, always ensure you correctly identify the faces and their corresponding shapes to accurately calculate the surface area.
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Math Problem Analysis
Mathematical Concepts
Pythagorean Theorem
Surface Area Calculation
Formulas
Pythagorean theorem: a^2 + b^2 = c^2
Surface area of a triangular face: 1/2 * base * height
Theorems
Pythagorean theorem
Suitable Grade Level
Grades 7-9
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