Math Problem Statement
Solution
Let's break down the problem to find the area of the rectangle step-by-step.
Step 1: Understanding the Geometry
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The problem provides:
- Two smaller identical circles, each with a radius of 6 cm.
- A larger circle that fits snugly above the two smaller circles.
- All three circles touch the sides of the rectangle.
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Diameter of Smaller Circles:
- Since each smaller circle has a radius of 6 cm, the diameter = cm.
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Width of the Rectangle:
- The width of the rectangle is equal to the combined diameters of the two smaller circles.
- Therefore, the width = cm.
Step 2: Finding the Height of the Rectangle
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The height of the rectangle consists of:
- The diameter of the larger circle.
- The radius of the smaller circles (since the larger circle sits directly on top of the smaller circles).
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Finding the Diameter of the Larger Circle:
- Since the large circle touches the two smaller circles, its diameter equals twice the sum of the smaller circle's radius.
- The radius of the larger circle can be derived using geometric properties, but for simplicity, let's consider that the arrangement is symmetric:
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Total Height of the Rectangle:
- Diameter of the larger circle (24 cm) + radius of the smaller circle (6 cm) = cm.
Step 3: Calculating the Area of the Rectangle
- Area Formula:
Final Answer
The area of the rectangle is 720.0 cm².
Would you like a detailed explanation of any step? Here are some follow-up questions you might consider:
- How do the properties of circles help determine their placement in the rectangle?
- Why is the arrangement of the circles inside the rectangle symmetrical?
- How would the problem change if the smaller circles had a different radius?
- Can this method be generalized for different sizes of circles in similar configurations?
- What geometric principles are used to derive the diameter of the larger circle?
Tip: When working with problems involving touching circles, consider symmetry and the relationships between radii and diameters to simplify calculations.
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Math Problem Analysis
Mathematical Concepts
Geometry
Area
Circles
Formulas
Area of a rectangle = width × height
Diameter of a circle = 2 × radius
Theorems
Properties of circles (touching circles and diameter relation)
Symmetry of circle arrangement in rectangles
Suitable Grade Level
Grades 8-10