Geometry and proof · Ages 13–14
Equilateral triangles, squares and hexagons worksheet
Part of: Polygons
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Equilateral triangles, squares and hexagons worksheet
Geometry and proof · Ages 13–14
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.A square ABCD is drawn on a circle of radius 5 cm, as shown; its diagonals AC and BD are both diameters. The diagonal AC =
Answer:
2.Draw a square ABCD, its vertices on a circle of radius 7 cm, with compasses, a straight edge and a set square: put the steps in order.
- A. AB = BC = CD = DA: join A, B, C and D in turn
- B. OA = OC = 7 cm: draw a circle, centre O and radius 7 cm, and a diameter AC
- C. ∠AOB = 90°: with a set square, draw the diameter BD at right angles to AC through O
Write the letters in the right order
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3.A regular hexagon is drawn in a circle of radius 5 cm, as shown. Its perimeter =
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4.Draw an equilateral triangle ABC of side 4 cm with compasses and a straight edge: put the steps in order.
- A. BC = AB: with centre B and the same radius, draw an arc crossing the first at C
- B. AB = BC = CA: join AC and BC
- C. AC = AB: with centre A and radius 4 cm, draw an arc
- D. AB = 4 cm: draw the line segment AB with the straight edge
Write the letters in the right order
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5.Draw a regular hexagon ABCDEF, its vertices on a circle of radius 5 cm, with compasses and a straight edge: put the steps in order.
- A. BC = CD = DE = EF = 5 cm: with centres B, C, D and E in turn, mark off C, D, E and F
- B. OA = 5 cm: draw a circle, centre O and radius 5 cm, and take a point A on it
- C. FA = 5 cm: join AB, BC, CD, DE, EF and FA
- D. AB = 5 cm: with the compasses open at 5 cm and centre A, mark off B on the circle
Write the letters in the right order
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6.A square ABCD is drawn on a circle of radius 12 cm, as shown; its diagonals are diameters at right angles. The area of ABCD =
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7.An arc with centre A and radius AB, and another with centre B and the same radius, cross at C. AB = 5 cm. The perimeter of △ABC =
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8.A regular hexagon ABCDEF is drawn in a circle of radius 6 cm, centre O. OA = cm, AB = cm, ∠AOB = °.
Fill in each blank
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9.A regular hexagon ABCDEF is drawn in a circle of radius 5 cm, centre O. OA = cm, AB = cm, ∠AOB = °.
Fill in each blank
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10.A regular hexagon ABCDEF has sides of 10 cm. The diagonal AD =
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Part B: Choose, or true or false (2 marks each)
11.On a circle of radius 8 cm, marking off 8 cm with the compasses comes back to the start after exactly 6 marks. Why?
- A. A hexagon's angles add to 720° = 2 × 360°
- B. OA = OB = AB = 8 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
- C. The circumference is exactly 6 times the radius
- D. A regular hexagon has 6 lines of symmetry
Choose the right answer
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12.On a circle of radius 5 cm, marking off 7 cm with the compasses comes back to the start after 6 marks, making a regular hexagon.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.A regular hexagon ABCDEF has sides of 8 cm and centre O. O is 8 cm from every vertex.
True or false?
Working:
Answer:
14.To draw an equilateral triangle ACE in a circle of radius 4 cm, draw a regular hexagon ABCDEF first, then join every other point: put the steps in order.
- A. AB = BC = CD = DE = EF = 4 cm: with the compasses open at 4 cm, mark off B, C, D, E and F round the circle from A
- B. OA = 4 cm: draw a circle, centre O and radius 4 cm, and take a point A on it
- C. AC = CE = EA: join A, C and E
Write the letters in the right order
Working:
Answer:
15.A regular hexagon ABCDEF is drawn in a circle of radius 5 cm, centre O. The perimeter of △OAB = cm, ∠ABC = °.
Fill in each blank
Working:
Answer:
16.To draw an equilateral triangle ABC (AB = 5 cm), arcs of radius AB are drawn with centres A and B, crossing at C. Why must △ABC be equilateral?
- A. AC = AB = 5 cm (one radius) and BC = AB = 5 cm (one radius): three equal sides
- B. AC = BC, so △ABC is isosceles
- C. ∠A and ∠B measure 60°
- D. C lies straight above AB's midpoint
Choose the right answer
Working:
Answer:
17.A round tablecloth has a radius of 10 cm. A trim is sewn along a regular hexagon whose corners lie on its edge. How long is the trim?
Working:
Answer:
18.A regular hexagon ABCDEF has sides of 12 cm and centre O. O is 13 cm from every vertex.
True or false?
Working:
Answer:
Show the answers and working
Equilateral triangles, squares and hexagons worksheet: Answers and working
Part A: Working out and filling in
1. 10 cm
AC is a diameter, two radii: 2 × 5 = 10 cm.
2. B → C → A
The circle and the diameter AC first, then the right angle at O for the diameter BD, then the four points joined. OA = OB = OC = OD = 7 cm and the four angles at O are 90°: a quarter turn about O carries each corner onto the next, so the four sides and the four angles are equal, and ABCD is a square.
3. 30 cm
A regular hexagon's side equals the circle's radius, 5 cm: perimeter = 6 × 5 = 30 cm.
4. D → C → A → B
The base AB first, then an arc of the same radius from each of A and B to find C, then join: AC and BC both equal the radius 4 cm, so all three sides are equal.
5. B → D → A → C
Open at the radius, each mark turns 60° at the centre; six marks are exactly round, back to A.
6. 288 cm²
The square is 4 right-angled triangles OAB, OBC, OCD and ODA, each 12 × 12 ÷ 2 = 72 cm², so 4 × 72 = 288 cm². Another way: diagonal × diagonal ÷ 2 = 24 × 24 ÷ 2 = 288 cm².
7. 15 cm
AC and BC are both the radius, equal to AB: three sides of 5 cm, perimeter 3 × 5 = 15 cm.
8. 6, 6, 60
OA is the radius, 6 cm; B was marked off at the radius, so AB = 6 cm; △OAB is equilateral, so ∠AOB = 60°.
9. 5, 5, 60
OA is the radius, 5 cm; B was marked off at the radius, so AB = 5 cm; △OAB is equilateral, so ∠AOB = 60°.
10. 20 cm
AD passes through the centre O: OA and OD are both the radius, equal to the side, 10 cm, so AD = 2 × 10 = 20 cm.
Part B: Choose, or true or false
11. B. OA = OB = AB = 8 cm, so △OAB is equilateral, the angle at the centre is 60°, and 6 × 60° = 360°
Each mark makes a triangle with O whose sides are all 8 cm: equilateral, 60° at the centre, and 6 of those make 360°.
12. False
Only when the compasses open at the radius does each mark turn 60° at the centre, 6 making a full turn. Here the radius is 5 cm and the opening 7 cm.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. True
A regular hexagon is six equilateral triangles round its centre: O is as far from each vertex as a side is long, 8 cm.
14. B → A → C
The circle first, then the hexagon's six points, then every other one joined: each side spans two equal steps, so the three are equal.
15. 15, 120
△OAB has three sides of 5 cm, perimeter 3 × 5 = 15 cm; ∠ABC is two equilateral triangles' 60°: 120°.
16. A. AC = AB = 5 cm (one radius) and BC = AB = 5 cm (one radius): three equal sides
Both arcs have radius AB: AC = AB and BC = AB, so all three sides are 5 cm.
17. 60 cm
The hexagon's side equals the radius, 10 cm: trim = 6 × 10 = 60 cm.
18. False
A regular hexagon is six equilateral triangles round its centre: O is as far from each vertex as a side is long, 12 cm.
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