No protractor is allowed in the test, so angle questions are never about measuring. Every one of them is solved by calculation from a small number of facts.
That is worth saying to a child directly, because a diagram that looks like it wants measuring is a diagram designed to mislead.
Remember what is not allowed in the room: no calculator, no ruler, no protractor, and no scrap paper, though working out in the question booklet is permitted.
No protractor, and why that matters
Diagrams in these questions are frequently not drawn to scale. An angle drawn looking like 90 degrees may be labelled 70, and the label is what counts.
So the instruction is simple: never judge an angle by eye. Read the labels, apply a rule, calculate. If a child is squinting at the page, they have already gone wrong.
The four facts
| Fact | Adds up to |
|---|---|
| Angles on a straight line | 180 degrees |
| Angles around a point | 360 degrees |
| Angles in a triangle | 180 degrees |
| Angles in a quadrilateral | 360 degrees |
Plus one more that is not a sum: vertically opposite angles are equal. Where two straight lines cross, the angles facing each other match.
Those five between them solve the overwhelming majority of angle questions at this level. They are worth having on a formula sheet and memorised cold.
Three worked examples
A triangle has angles of 65 and 48 degrees. What is the third?
65 plus 48 is 113. A triangle totals 180, so the third angle is 180 minus 113, which is 67 degrees.
Two angles sit on a straight line. One is 125 degrees. What is the other?
180 minus 125 is 55 degrees.
What is each interior angle of a regular hexagon?
A hexagon splits into 4 triangles, so its angles total 4 times 180, which is 720. Regular means all equal, so each is 720 divided by 6, which is 120 degrees.
That last method generalises: a shape with n sides splits into n minus 2 triangles.
Shape properties worth knowing
- Equilateral triangle: all sides equal, all angles 60.
- Isosceles triangle: two equal sides, and the two base angles are equal.
- Right angle: exactly 90. Acute is less, obtuse is more than 90 but less than 180, reflex is more than 180.
- Square: four equal sides, four right angles, four lines of symmetry.
- Rectangle: four right angles, two lines of symmetry.
- Parallelogram: opposite sides parallel and equal, opposite angles equal.
The isosceles fact is the one that unlocks the most questions. Spotting that two sides are marked equal tells you two angles are equal, which usually turns a question with two unknowns into one with a single unknown.
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If they are squinting at the diagram, they have gone wrong
The single most useful thing to tell a child about angles in this test is that there is no protractor in the room. Nothing is measured. Everything is worked out.
That matters because the diagrams are often not drawn to scale, and some of them are drawn to mislead. An angle that looks like a right angle may be labelled seventy. The label is the fact; the picture is just a picture.
This is also a topic where the formula sheet earns its keep. Four sums and the vertically opposite rule. We give students a mind map to memorise exactly so that recall is instant, because the speed of answering is what causes most of the mistakes, not the difficulty of the mathematics.
Mushtaque Ahmed, Founder of GrammarPath and Bright Mentors
Frequently asked questions
Is a protractor allowed in the 11+?
No. The partnership's familiarisation guide lists the protractor alongside the ruler and calculator as not permitted. That means every angle question must be solved by calculation rather than measurement.
Are 11+ angle diagrams drawn to scale?
Frequently not, and that is deliberate. An angle drawn looking like a right angle may be labelled 70 degrees, and the label is what counts. A child judging by eye is being set up to fail.
What angle facts does my child need for the 11+?
Four sums and one equality: angles on a straight line total 180, around a point 360, in a triangle 180 and in a quadrilateral 360, and vertically opposite angles are equal. Those five solve the large majority of questions at this level.
How do you find the interior angle of a regular polygon?
Split it into triangles: a shape with n sides splits into n minus 2. A hexagon gives 4 triangles, so 4 times 180 is 720 degrees in total, and dividing by 6 gives 120 degrees for each angle.
Which shape property unlocks the most questions?
The isosceles triangle. Two equal sides means two equal base angles, which usually converts a problem with two unknowns into one with a single unknown.