The quarter-turn segment

At 90°, the triangle and the segment have the simplest numbers.

The rule

At 90° and radius 7, the areas are 38.5, 24.5 and 14.

Sector = 154 ÷ 4 = 38.5. Triangle = ½ × 7 × 7 = 24.5. Segment = 38.5 − 24.5 = 14 square units.

A segment's edge is an arc plus a chord, not two radii.

The edge of a shape is its length all the way round. The whole rim is 44. The arc is a quarter of it: 44 ÷ 4 = 11. The chord is √(49 + 49) = 7√2. The edge = 11 + 7√2.

You can check the area by counting cells. A cell is one small square of a grid. Place a 7-by-7 grid over the quarter disc. A cell counts when its centre is inside the shape. A centre on the chord goes to the triangle. Curved and slanted edges cut some cells into parts. So counting only estimates. The counts are 39, 28 and 11. They count the sector, the triangle and the segment in order. The exact areas are 38.5, 24.5 and 14.

Tap each region. Count its shaded squares.

Try the lab to finish this lesson.