How To Really Solve This Fun Geometry Puzzle? (II) - Whiteboard graphics showing a circle two radii that are perpendicular to each other. Along these radii lie the sides of a rectangle with length 4 units and breadth 3 units respectively. The area within this construction that is not occupied by the rectangle is shaded yellow. Shaded area = What??

Welcome to the second entry in the fun geometry puzzle series. I recently encountered this geometry puzzle from multiple sources. Although it is on the easy side, it managed to engage me and I had fun solving it. So, I thought I’d write about it.

We have a circle with the centre ‘O’. From this centre, we have two perpendicular line segments drawn along the radii that meet the circle at its periphery. 

Inside this construction, we have a rectangle with a length of 4 units along the horizontal line segment and a breadth of 3 units along the vertical line segment.

Given this setting, your task is compute the area that is not occupied by the rectangle in the construction. This region is shaded in yellow. How fast can you solve this puzzle?

Spoiler Alert

If you wish to give this puzzle a try on your own, I suggest that you pause reading this essay at this point and go ahead with your attempt. After your attempt, you may continue reading.

Beyond this section, I will be explicitly discussing the solution to this puzzle.

This essay is supported by Generatebg

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Setting Up the Fun Geometry Puzzle

It is clear that the rectangle is going to play a key role in this puzzle. So, let us begin by labelling the vertices of the rectangle. I chose to mark them in the anticlockwise direction as ‘OABC’ like so:

How To Really Solve This Fun Geometry Puzzle? (II) — The same whiteboard graphics from before is presented again. But this time, the vertices of the rectangle are marked as O, A, B, and C in the anticlockwise direction. Note that O also marks the centre of the circle.
Labelled Rectangle — Illustration created by the author (geometry not to scale)

Our next challenge is to somehow relate the rectangle with the circle. Currently, we know that a weak link exists, in that the sides of the rectangle lie along the radii of the circle. So, how shall we compute the radius of the circle from this information?

The answer: the Pythagorean theorem. We can use the sides ‘OA’ and ‘OC’ to compute the diagonal ‘OB’ using the Pythagorean theorem. ‘OB’ then gives us the radius of the circle.

How To Really Solve This Fun Geometry Puzzle? (II) — The labelled whiteboard illustration is presented on the left. On the right, the following piece of math is illustrated: By applying the pythagorean theorem: OB = √[(OA)² + (OB)²] = √(⁴² + ³²) = √(25) = 5 units.
Length of diagonal — Illustration created by the author (geometry not to scale)

So, it turns out that the radius of our circle is 5 units. We can now use this information to solve the rest of the puzzle.

The Solution to the Fun Geometry Puzzle

The sides of a rectangle are perpendicular to each other. So, the radii that lie along the sides of our rectangles are also at right angles to each other. What this means is that the construction we are interested in is a quadrant; a circle can be divided into 4 quadrants. 

So, the area of the entire quadrant is one-fourth of the circle’s area (which we can compute using the radius we have now). We have only one final step left.

We can compute the area of the shaded portion by subtracting the area of the rectangle (length×breadth) from the area of the quadrant of the circle as follows:

Shaded Area = (Area of quadrant)−(Area of rectangle) → This information is graphically illustrated using a circle whose right-hand top quadrant is shaded yellow and a rectangle on the right which is also shaded yellow. The shaded area is obtained by subtracting the area of the rectangle from the quadrant area. Therefore, shaded area = [(π×r²)/4]−[Length×Breadth] = [(π×25)/4]−[4×3] = 7.635 units² (approximately).
The solution — Illustration created by the author (geometry not to scale)

There we go. The area of the shaded portion is approximately 7.635 units². I hope you enjoyed solving this geometry puzzle as much as I did. Keep an eye on this space for more fun geometry puzzles in the future!

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