Question

Here is a diagram of a black circle inside a white square and a diagram of a small grey square inside a white square.

Answer 1

The ratio of the area of the black circle to the area of the white square to the area of the gray square is π : 4 : 2

How to determine the ratio of the areas?

The side length AB is given as:

AB = 8x

The area of the white square is calculated as

White = (AB)²

This gives

White = (8x)²

Evaluate

White = 64x²

The side length AB represents the diameter of the black circle.

So, the radius is:

r =AB/2

This gives

r = 8x/2 = 4x

The area is then calculated as:

Circle = πr²

This gives

Circle = π(4x)²

Evaluate

Circle = 16πx²

Next, calculate AW and AZ using:

AZ = AW = AB/2

Evaluate

AZ = AW = 8x/2 = 4x

Calculate WZ using the following Pythagoras theorem

WZ² = AW² + AZ²

This gives

WZ² = (4x)² + (4x)²

Evaluate

WZ² = 2(4x)²

Take the square root of both sides

WZ = 4x√2

The area of the gray square is:

Gray = WZ²

This gives

Gray = 2(4x)²

Evaluate

Gray = 32x²

At this point, the areas are:

• White = 64x²
• Circle = 16πx²
• Gray = 32x²

The ratio is then represented as:

Ratio = Circle : White : Gray

This gives

Ratio = 16πx² : 64x² : 32x²

Divide through by 16x²

Ratio = π : 4 : 2

Hence, the ratio of the area of the black circle to the area of the white square to the area of the gray square is π : 4 : 2

Read more about areas at:

brainly.com/question/24487155

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