Convex & Concave Polygons
Every polygon is either convex or concave, and one look at its largest interior angle is enough to decide which.
Convex polygon
A polygon is convex if the measure of every interior angle is less than 180° — it contains no reflex interior angle.
Straight-line test: draw a line through any two consecutive vertices — all remaining vertices stay on one side of it.
Concave polygon
A polygon is concave if at least one interior angle measures more than 180° — it contains at least one reflex interior angle.
Straight-line test: some line through two consecutive vertices leaves the remaining vertices on two different sides.
Example — deciding from the angles
In quadrilateral ABCD: m(∠B) = m(∠D) = 5X°, m(∠A) = 12X°, m(∠C) = 50X°. Find X and state whether the shape is convex or concave.
∵ ABCD is a quadrilateral
∴ sum of interior angles = 360°
∴ 12X + 5X + 50X + 5X = 360
∴ 72X = 360 ⟹ X = 5
∴ m(∠C) = 50 × 5 = 250° (a reflex angle)
∴ ABCD is a concave polygon.
Try it yourself
In quadrilateral ABCD: m(∠A) = 6X°, m(∠B) = 7X°, m(∠C) = 12X°, m(∠D) = 11X°. Find X, then decide whether the shape is convex or concave.
Answer: X = 10, largest angle = 120° < 180° ⟹ convex.