Grade 4 Polygons and Diagonals

What this problem type covers

Twenty problem types across five levels: the diagonals of a regular polygon, the angles of a polygon added, a wire bent from one regular shape into another, a square inside a circle inside a square, two diagonal counts added, a hexagon tiled with trapezoids, the angle diagonals leave at a vertex, two regular shapes joined edge to edge, a polygon named by its angle sum, a rhombus whose diagonals make an equilateral triangle, a triangle inside a rhombus from the two diagonals, a rectangle joined to an equilateral triangle, a pentagon joined to a rhombus, a shape measured by the pieces it is made from, five angles found by drawing in a pentagon, a polygon named by its diagonal count, a run of hexagons counted from its perimeter, a regular polygon found from the widest angle two diagonals make, the shapes a number of squares can make, and the piece left out of a shape.

Example problems

One example from each difficulty band, with worked solutions.

These problems are generated automatically. The answers are derived as each problem is built and are checked by tests, but mistakes are still possible — please tell us if you find one.

1. 1 · ★1
   What do all the angles of a regular polygon with 12 sides add to?
   
   • ①2,160 degrees
   • ②360 degrees
   • ③1,980 degrees
   • ④1,800 degrees
   • ⑤150 degrees
   Show solution

   ④1,800 degrees

   1. Diagonals from one vertex split a regular polygon with 12 sides into 10 triangles.
   2. The angles of one triangle add to 180 degrees.
   3. So they all add to 180 × 10 = 1,800 degrees.
2. 2 · ★2
   A circle is drawn inside a square of side 52 cm, and four points on the circle make the square ㄱㄴㄷㄹ. How long is ㄱㅇ?
   
   • ①208 cm
   • ②52 cm
   • ③104 cm
   • ④26 cm
   • ⑤13 cm
   Show solution

   ④26 cm

   1. The circle's diameter equals the outer square's side, 52 cm.
   2. That diameter is the diagonal of the inner square.
   3. In a square each diagonal cuts the other in half.
   4. So ㄱㅇ is 52 ÷ 2 = 26 cm.
3. 3 · ★3
   Diagonals are drawn in the rhombus ㄱㄴㄷㄹ. The marked angle is 30 degrees and ㅇㄷ is 10 cm. What do the three sides of triangle ㄱㄴㄷ add to?
   
   • ①30 cm
   • ②10 cm
   • ③20 cm
   • ④60 cm
   • ⑤40 cm
   Show solution

   ④60 cm

   1. The diagonals of a rhombus cross at right angles.
   2. The marked 30 degrees leaves 60, which makes ㄱㄴㄷ equilateral.
   3. Each diagonal halves the other, so ㄱㄷ is 10 + 10 = 20 cm.
   4. So the three sides add to 20 × 3 = 60 cm.
4. 4 · ★4
   The two diagonals of the rhombus ㄱㄴㄷㄹ add to 40 cm and differ by 24 cm. Its sides are 9 cm. What is the perimeter of triangle ㄱㄴㅇ?
   
   • ①29 cm
   • ②49 cm
   • ③27 cm
   • ④40 cm
   • ⑤20 cm
   Show solution

   ①29 cm

   1. The sum and difference give diagonals of 32 cm and 8 cm.
   2. Each halves the other, so the halves are 16 cm and 4 cm.
   3. So the perimeter is 9 + 16 + 4 = 29 cm.
5. 5 · ★5
   There is one piece of each of the sizes 8, 5, 6, 5. All but one were used to make a shape of size 19. How big is the piece that was left out?
   
   • ①19
   • ②24
   • ③4
   • ④6
   • ⑤5
   Show solution

   ⑤5

   1. The pieces add up to 24.
   2. The shape made comes to 19.
   3. So the piece left out is 24 − 19 = 5.

Where people usually go wrong

• Forgetting to halve after counting the diagonals from every vertex.
• Splitting a polygon into n triangles rather than n − 2.
• Doubling the side when the pieces needed grow with its square.
• Using the whole diagonals of a rhombus where only their halves are wanted.
• Adding to 180 where a full turn of 360 is needed.
• Counting turns and flips of the same shape as different shapes.
• Giving an angle of a polygon as more than 180 degrees.
• Answering with the number of sides when the perimeter was asked for.