Polygons · Ages 13–14
Drawing a regular pentagon
72° at the centre, 108° inside; mark the side CN off five times.
How to do it
Method
72° at the centre, 108° inside; mark the side CN off five times
∠BAC = (180° − 108°) ÷ 2 = 36°
Another way
By the exterior angles
360° ÷ 5 = 72°, 180° − 72° = 108°
In full
A regular pentagon's angle at the centre is 360° ÷ 5 = 72°, and each interior angle is (5 − 2) × 180° ÷ 5 = 108°. One way: draw a circle, centre O, and two diameters at right angles, AB and CD (the right angle with a set square or a protractor); measure OB with a ruler to find its midpoint M; with centre M and radius MC, draw an arc to cut OA at N; CN is the pentagon's side, so with centre C and radius CN mark it off round the circle five times, and join the points in turn. Then check: every angle at the centre should be 72°. Its diagonals give many more angles: △ABC is isosceles (AB = BC) with ∠B = 108°, so ∠BAC = (180° − 108°) ÷ 2 = 36°. Another way to the angle: by the exterior angles, 360° ÷ 5 = 72° each, so interior = 180° − 72° = 108°.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
O is the centre of the regular pentagon. x =
Show the answer and working
Answer: 72°
The five angles at the centre are equal and make a full turn: x = 360° ÷ 5 = 72°.
Example 2
Each angle of a regular pentagon is 108°.
True or false?
Show the answer and working
Answer: True
The angles add to (5 − 2) × 180° = 540°, each 540° ÷ 5 = 108°; 72° is its angle at the centre and its exterior angle.
Example 3
The regular pentagon ABCDE has diagonals AC and AD. ∠BAC = °, ∠CAD = °, ∠DAE = °.
Fill in each blank
Show the answer and working
Answer: 36, 36, 36
∠BAC = ∠DAE = (180° − 108°) ÷ 2 = 36° (base ∠s, isos. △), so ∠CAD = 108° − 36° − 36° = 36°: the diagonals cut 108° into three equal parts.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.O is the centre of the regular pentagon. x =
Answer:
2.Draw a regular pentagon in a circle of radius 4 cm with compasses, a ruler and a set square: put the steps in order.
- A. ∠AOC = 90°: draw a circle, centre O and radius 4 cm, and with a set square two diameters AB and CD at right angles
- B. side = CN: with centre C and radius CN, mark off round the circle five times and join the points in turn
- C. OM = MB: measure OB with the ruler to find its midpoint M
- D. MN = MC: with centre M and radius MC, draw an arc cutting OA at N
Write the letters in the right order
Answer:
3.ABCDE is a regular pentagon and AC a diagonal. ∠BAC =
Answer:
4.Each angle of a regular pentagon is 72°.
True or false?
Answer:
5.The regular pentagon ABCDE has diagonals AC and AD. ∠BAC = °, ∠CAD = °, ∠DAE = °.
Fill in each blank
Answer:
All the answers
- 1. 72°
- 2. A → C → D → B
- 3. 36°
- 4. False
- 5. 36, 36, 36
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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