TMUA: Geometry and vectors
Geometry: angles, polygons, congruence, similarity, areas and volumes.
- Section 1 Part 2 · M1-M7 GCSE-level knowledge (Part 2) · spec M5
- Papers 1 and 2
- 7 practice questions
- No calculator
What the specification covers
- Angle facts, polygons, congruence and similarity
- Circle theorems, areas and volumes (sphere, cone and pyramid formulae given if needed)
- Vectors in geometric arguments
Key ideas
- Exterior angles of a polygon add to \(360^\circ\); interior angle \(=180^\circ-\) exterior angle.
- An \(n\)-gon has \(\tfrac{n(n-3)}{2}\) diagonals.
- Similar shapes with length scale factor \(k\) have area factor \(k^2\) and volume factor \(k^3\).
- Formulae for spheres, cones and pyramids are given if needed; circle and cylinder formulae are not.
Common mistakes
- Congruence by SSA is not a valid test (except RHS).
- Arc length uses the angle as a fraction of \(360^\circ\) (or radians).
Exam tip
Find the number of sides from the exterior angle first.
Worked example
Worked example
Each interior angle of a regular polygon is \(156^\circ\). How many diagonals does the polygon have?
- \(180\)
- \(105\)
- \(75\)
- \(90\)
- \(78\)
Answer: D: \(90\)
Each exterior angle is \(24^\circ\), so there are \(360/24=15\) sides. Each vertex joins to 12 non-adjacent vertices: \(\tfrac{15\times12}{2}=90\) diagonals.
Practice questions
Try each question before opening the solution. No calculator: the TMUA does not allow one. Every answer here was re-checked independently by computer before it was published.
Question 1
Each interior angle of a regular polygon is \(135^\circ\). How many diagonals does the polygon have?
- \(8\)
- \(20\)
- \(40\)
- \(28\)
- \(24\)
Show the answer and solution
Answer: B: \(20\)
Each exterior angle is \(45^\circ\), so there are \(360/45=8\) sides. Each vertex joins to \(5\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{8\times5}{2}=20\).
Question 2
Each interior angle of a regular polygon is \(165^\circ\). How many diagonals does the polygon have?
- \(252\)
- \(24\)
- \(264\)
- \(276\)
- \(504\)
Show the answer and solution
Answer: A: \(252\)
Each exterior angle is \(15^\circ\), so there are \(360/15=24\) sides. Each vertex joins to \(21\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{24\times21}{2}=252\).
Question 3
Each interior angle of a regular polygon is \(162^\circ\). How many diagonals does the polygon have?
- \(340\)
- \(180\)
- \(170\)
- \(20\)
- \(190\)
Show the answer and solution
Answer: C: \(170\)
Each exterior angle is \(18^\circ\), so there are \(360/18=20\) sides. Each vertex joins to \(17\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{20\times17}{2}=170\).
Question 4
Each interior angle of a regular polygon is \(160^\circ\). How many diagonals does the polygon have?
- \(18\)
- \(270\)
- \(153\)
- \(135\)
- \(144\)
Show the answer and solution
Answer: D: \(135\)
Each exterior angle is \(20^\circ\), so there are \(360/20=18\) sides. Each vertex joins to \(15\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{18\times15}{2}=135\).
Question 5
Each interior angle of a regular polygon is \(168^\circ\). How many diagonals does the polygon have?
- \(810\)
- \(30\)
- \(405\)
- \(420\)
- \(435\)
Show the answer and solution
Answer: C: \(405\)
Each exterior angle is \(12^\circ\), so there are \(360/12=30\) sides. Each vertex joins to \(27\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{30\times27}{2}=405\).
Question 6
Each interior angle of a regular polygon is \(140^\circ\). How many diagonals does the polygon have?
- \(54\)
- \(36\)
- \(9\)
- \(31\)
- \(27\)
Show the answer and solution
Answer: E: \(27\)
Each exterior angle is \(40^\circ\), so there are \(360/40=9\) sides. Each vertex joins to \(6\) non-adjacent vertices, and each diagonal is counted twice: \(\tfrac{9\times6}{2}=27\).
Keep going
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- Next topic: Statistics and probability
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