Shown below are three views of the same solid. Count the number of edges.
Show Hint
For edge-counting problems with complex 3D shapes, it is highly effective to categorize the edges (e.g., by orientation like horizontal/vertical, or by location like top/bottom/side). This structured approach minimizes the risk of missing edges or counting them more than once.
Step 1: Understanding the Concept:
An edge of a solid is a line segment where two of its faces meet. This question requires visualizing a 3D object from its 2D representations and then systematically counting all its edges, including those that might be hidden in a particular view.
Step 2: Detailed Explanation:
To count the edges accurately, we can categorize them based on their position and orientation on the solid. Let's break down the object into logical parts.
Curved Edges: The object has two prominent curved "arms". Each arm has one curved edge at the top.
Number of curved edges = 2.
Top Straight Edges: Between the two curved arms, there is a flat top surface. This surface has a straight edge at the front and a corresponding straight edge at the back.
Number of top straight edges = 2.
Vertical Edges: There are four vertical edges that form the "sides" of the arms. Two are on the outer sides, and two are on the inner sides.
Number of vertical edges = 4.
Bottom Edges: The base of the object consists of three sections (under the left arm, the middle, and the right arm). Each section has a front edge and a back edge.
- Bottom edges for the left arm (front and back) = 2.
- Bottom edges for the middle section (front and back) = 2.
- Bottom edges for the right arm (front and back) = 2.
Total number of bottom edges = \(2+2+2 = 6\).
Total Count:
To find the total number of edges, we sum the counts from each category:
\[ \text{Total Edges} = (\text{Curved}) + (\text{Top Straight}) + (\text{Vertical}) + (\text{Bottom}) \] \
\[ \text{Total Edges} = 2 + 2 + 4 + 6 = 14 \] \
Step 3: Final Answer:
By systematically counting, we find that the solid has a total of 14 edges.
