Mathematics • Year 7 • Unit 3 • Lesson 13
Polygon Angles, Real World
Honeycomb hexagons, stop-sign octagons, soccer-ball pentagons, paving tiles. Use the interior-sum and exterior-sum formulas to figure out the angles in real shapes around you.
1. Word problems
For each, name the polygon, apply the formula, show your working.
1.1, Honeycomb. Bee honeycomb cells are regular hexagons. What is the size of each interior angle of one honeycomb cell? 2 marks
1.2, Stop sign. A standard road STOP sign is a regular octagon. (a) What is the sum of its interior angles? (b) What is the size of each interior angle? 2 marks
1.3, Soccer ball. A traditional soccer ball is made of black regular pentagons and white regular hexagons stitched together. (a) Each interior angle of a regular pentagon = ? (b) Each interior angle of a regular hexagon = ? (c) At a vertex where ONE pentagon and TWO hexagons meet, what is the total of the three angles? (d) Is that less than, equal to, or greater than 360°? 3 marks
1.4, Paving tile. An architect designs a paving tile shaped like an irregular pentagon. Four of its interior angles measure 110°, 130°, 100° and 105°. Find the fifth interior angle. 2 marks
1.5, Designer's puzzle. A landscape designer wants to make a regular polygon-shaped flowerbed whose each interior angle is 162°. How many sides does the flowerbed have? 2 marks
2. Explain your thinking
Full sentences. 4 marks
2.1 A floor tiler is trying to tile a floor using only regular pentagons (5-sided regular polygons). She finds that no matter how she lays the tiles, there are always gaps at the vertices. In a short paragraph: (i) Calculate each interior angle of a regular pentagon. (ii) Calculate how many pentagons would meet at a vertex if there were NO gap (i.e. 360° at each vertex). (iii) Explain why the number you get is NOT a whole number, and what that means for tiling. (iv) Suggest one regular polygon that DOES tile the floor with no gaps, and explain why.
How did this worksheet feel?
What I'll revisit before next class:
1.1, Honeycomb (regular hexagon)
S = (6 − 2) × 180° = 720°. Each interior = 720° ÷ 6 = 120°.
1.2, STOP sign (regular octagon)
(a) S = (8 − 2) × 180° = 6 × 180° = 1080°.
(b) Each interior = 1080° ÷ 8 = 135°.
1.3, Soccer ball vertex
(a) Pentagon: (5 − 2) × 180° ÷ 5 = 540° ÷ 5 = 108°.
(b) Hexagon: (6 − 2) × 180° ÷ 6 = 720° ÷ 6 = 120°.
(c) Total at vertex = 108° + 120° + 120° = 348°.
(d) 348° is less than 360°. The missing 12° forces the surface to curve into a ball, it can't lie flat.
1.4, Irregular pentagon (110, 130, 100, 105, x)
Sum = (5 − 2) × 180° = 540°. 110 + 130 + 100 + 105 + x = 540 → 445 + x = 540 → x = 95°.
1.5, Flowerbed, each interior 162°
Exterior = 180° − 162° = 18°. n = 360° ÷ 18° = 20 sides (a regular icosagon).
2.1, Why regular pentagons don't tile (sample response)
Each interior angle of a regular pentagon is (5 − 2) × 180° ÷ 5 = 108°. For tiles to fit around a single vertex with NO gap, the angles meeting at that vertex must sum to exactly 360°. The number of pentagons needed = 360° ÷ 108° = 3.333…, which is NOT a whole number. Because you can't have a fraction of a tile meet at the vertex, three pentagons leave a gap (3 × 108 = 324°, short by 36°) and four pentagons overlap (4 × 108 = 432°). So regular pentagons cannot tile a flat floor. A regular polygon that DOES tile is the regular hexagon: each interior angle is 120°, and 360 ÷ 120 = 3 exactly, three hexagons meet perfectly at every vertex (like a honeycomb). Equilateral triangles (60°, 6 per vertex) and squares (90°, 4 per vertex) also tile.
Marking: 1 for 108°; 1 for 360 ÷ 108 = 3.33…; 1 for explaining the non-integer = no tiling; 1 for naming a polygon that does tile with valid reason.