Geometry Puzzle – Challenge 66

Challenge:

What is the degree between the hour hand and minute hand of a clock at 4:30?

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From question 64 we learned that an hour hand of a clock moves 30 degree in one hour. At 4:30, the hour hand of the clock has moved 135 degree. (4.5 × 30 = 135)
At 4:30, the minute hand has moved 180 degree. So, the angle between hour and minute hand of the clock equals:
180 – 135 = 45 degree

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Frequently Asked Questions

How many degrees does each hour mark cover on a clock?

360 degrees divided by 12 hour marks = 30 degrees per mark. The angle from 12 to 1 is 30 degrees, from 12 to 2 is 60 degrees, and so on.

How fast does the minute hand move?

The minute hand sweeps 360 degrees every 60 minutes — that is 6 degrees per minute. So at any minute m past the hour, the minute hand sits at 6m degrees clockwise from 12.

How fast does the hour hand move?

The hour hand sweeps 360 degrees every 12 hours, so 30 degrees per hour, or 0.5 degree per minute. By 4:30, the hour hand has moved 4.5 hours times 30 = 135 degrees from 12.

What is the general formula?

At time h:m (with h in 1-12 and m in 0-59), the angle is |30h – 5.5m| degrees, taking the smaller of that value and 360 minus that value. For 4:30: |30(4) – 5.5(30)| = |120 – 165| = 45 degrees.

Why does the hour hand keep moving between the hours?

Because the hour hand smoothly sweeps — it does not jump from 4 to 5 instantly at 5:00. At 4:30, it has moved half-way from 4 to 5, which is 15 degrees beyond the 4 mark. So its total displacement from 12 is 120 + 15 = 135 degrees.

Walk through 3:15.

Minute hand at 15 minutes = 6(15) = 90 degrees. Hour hand at 3 plus a quarter of the way to 4: 30(3) + 0.5(15) = 90 + 7.5 = 97.5 degrees. Difference = 97.5 – 90 = 7.5 degrees.

When are the hour and minute hands exactly aligned?

Set the angles equal: 30h + 0.5m = 6m, so 30h = 5.5m, giving m = 60h/11. The hands align about every 65.45 minutes — eleven times in 12 hours.

When do the hands form a right angle?

When the angle between them is 90 degrees. Solve |30h – 5.5m| = 90 (or 270, since 360 – 90 = 270). Twenty-two times in a 12-hour cycle.

Why is the angle at 6:00 exactly 180 degrees?

Because the minute hand is at the 12 position (0 degrees) and the hour hand is exactly at the 6 position (180 degrees). The hands are diametrically opposite, forming a straight line.

What real-world skills does this puzzle build?

Understanding angular motion (the same math powers gear design, navigation, and any rotating system), unit conversion (degrees, minutes, hours), and translating a real situation into algebra. Clock problems show up in standardized tests across grade levels precisely because they exercise so much at once.

Related Lessons You May Like

• How to find the area of rectangles
• How to find the perimeter of rectangles
• How to find complementary, supplementary, vertical, and adjacent angles
• How to solve angles and angle measure
• How to solve multi-step word problems

If your student enjoys puzzles like this, Geometry for Beginners dives into the same kinds of relationships inside a full curriculum. For the algebra you will lean on, Pre-Algebra for Beginners fills in the foundations gently.