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.

Te zadania są generowane automatycznie. Odpowiedzi powstają wraz z zadaniem i są sprawdzane testami, ale błędy są możliwe — daj nam znać, jeśli jakiś znajdziesz.

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.