Math Problem Statement

Fill in the missing numbers to complete the mapping for the given circle's radius and area.

Solution

The image shows a table where the radius of a circle is mapped to its area. The relationship between the radius rr and the area AA of a circle is given by the formula:

A=πr2A = \pi r^2

We are provided with areas in the form of π\pi multiples (like 9π9\pi, 36π36\pi, and 144π144\pi), and we need to find the corresponding radii xx, yy, and zz.

Solving for xx, yy, and zz:

  1. For 9π9\pi area: A=πr29π=πr2r2=9r=3A = \pi r^2 \quad \Rightarrow \quad 9\pi = \pi r^2 \quad \Rightarrow \quad r^2 = 9 \quad \Rightarrow \quad r = 3 So, x=3x = 3.

  2. For 36π36\pi area: A=πr236π=πr2r2=36r=6A = \pi r^2 \quad \Rightarrow \quad 36\pi = \pi r^2 \quad \Rightarrow \quad r^2 = 36 \quad \Rightarrow \quad r = 6 So, y=6y = 6.

  3. For 144π144\pi area: A=πr2144π=πr2r2=144r=12A = \pi r^2 \quad \Rightarrow \quad 144\pi = \pi r^2 \quad \Rightarrow \quad r^2 = 144 \quad \Rightarrow \quad r = 12 So, z=12z = 12.

Final values:

  • x=3x = 3
  • y=6y = 6
  • z=12z = 12

Would you like further details on the calculations or have any questions?

Here are 5 related questions to deepen understanding:

  1. What is the formula for the circumference of a circle?
  2. How would you calculate the radius if the area is given as 25π25\pi?
  3. What happens to the area if the radius of a circle is doubled?
  4. Can you find the diameter of the circle if the area is 9π9\pi?
  5. How is the area of a sector related to the total area of a circle?

Tip: Always check if units are consistent, especially when working with geometric shapes like circles. This ensures accurate calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Area of a Circle
Radius

Formulas

Area of a circle A = πr^2

Theorems

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Suitable Grade Level

Grades 6-8