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The prompt
The prompt
Mathematical inquiry processes: Generate examples and verify; generalise and prove. Conceptual field of inquiry: Inverse operations; area of a triangle; area of a sector and length of an arc.
Tariq Rasool, a teacher of mathematics in London (UK), designed the prompt after one of his year 10 students used the height of a right-angled triangle drawn on the radius to calculate the length of an arc (see 'Origin of the prompt' below).
The teacher can use the prompt to introduce students to the formulae for the area of a sector and the length of an arc.
At first, students are sceptical about the truth of the prompt. That it is always true normally comes as a surprise.
Question, notice, and wonder
Question, notice, and wonder
In the question, notice, and wonder phase of the inquiry students have made these contributions:
The triangle is a right-angled triangle.
The sector is part of a circle.
How do you work out the length of an arc?
Is the radius the same as the base of the triangle?
The statement is false because it depends on the angle at the centre of the sector.
We need to know the length of the radius and the angle to work out the area of the sector.
It is not possible for the length of the arc and the height of the triangle to be equal. The length is irrational because you use π in the calculation, whereas the height is rational.
What are the areas of the three regions in the diagram?
Before deciding on the direction of the inquiry, the teacher should ensure that the class understands the diagram. In particular, the base of the triangle lies on the radius and, therefore, the lengths of base and radius are the same. This is implied rather than being given explicitly, which it would have to be in a mathematically rigorous statement of the generalisation.
The teacher should also explain how to calculate the length of an arc, especially if the question arises in the initial phase of the inquiry. In a process of co-construction, the teacher draws on students' existing knowledge of the area and circumference of a circle and extends it to sectors.
Guided inquiry
Guided inquiry
In the slides there are structured lines of inquiry based on the six regulatory cards shown above. The teacher should guide students to select Practise a procedure or Find more examples to start the inquiry. When students have shown they can calculate the length of an arc given two pieces of information about the sector, they move onto finding connections, extending relationships, analysing structure, or proving the generalisation.
March 2026
Origin of the prompt
Origin of the prompt
Tariq designed the prompt after one of his year 10 students used the height of a right-angled triangle to solve a question about the perimeter of a sector (see picture).
Calculations
Calculations
The answer to the question requires three calculations (solutions at each step are rounded to 2 decimal places):
Work out the angle using the area and the length of the radius: 40 x 360 ÷ π(72) = 93.54o
Calculate the length of the arc: 93.54 ÷ 360 x 2π(7) = 11.43 cm
Add the lengths of the two radii to the length of the arc to find the perimeter: 2 x 7 + 11.43 = 25.43 cm
Instead of finding the size of the angle, the student attempted to create a simpler analogous problem. He sketched a right-angled triangle with the same area as the sector on the radius (see the picture). As the height of the triangle looked equal to the length of the arc, he used 40 = 0.5(7)h to calculate the height: h = 40 ÷ 0.5(7) = 11.43 cm
As 11.43 cm is also the length of the arc, the student went on to get the right answer to the question.
Particular or general?
Particular or general?
At first Tariq thought the result might be a coincidence that worked only for the particular case in the question. However, after exploring a few examples, he realised that the height of the triangle is always equal to the length of the arc when the areas of the triangle and the sector are equal.
After devising a proof, Tariq suggested that the generalisation would make an intriguing inquiry prompt.
The student who had assumed that areas and lengths are equal in his sketch did not know of the general result. Tariq could not find reference to it online. Please Contact Inquiry Maths if you have any information about the generalisation.
Lines of inquiry
Lines of inquiry
The lines of inquiry in the slides are not meant to be followed sequentially, although students should start with one of the first two to develop fluency in calculating the length of an arc.
Once the class has verified that the contention in the prompt is true in a number of cases and has examined counter-examples for calculation errors, it considers following other lines of inquiry.
The next three encourage students to look for connections and patterns related to the areas of the regions in the diagram. Individuals might decide, however, to jump directly to devising a proof of the generalisation.
In the slides, there are structured tasks and open-ended activities for each line of inquiry. Click on the arrows to see a description.
Students derive the angle and all lengths and areas in the diagram given two properties of the sector. Each pair of questions involves doubling and halving the properties. Students leave lengths and areas in exact form.
The task leads into an open-ended activity in which students consider what happens to the area of the sector and length of the arc if they double both properties. This line of inquiry culminates in making and explaining a generalisation.
Students generate their own examples by selecting an angle and an area for the sector. They show the length of the arc and the height of the triangle are equal in all cases.
The teacher might use the example in the slides to model the procedure.
The next line of inquiry involves a change to the diagram. Now both radii lie on sides of the right-angled triangle (see picture below). What is the area of the shaded region as a proportion of the area of the triangle?
Students explore what happens to the proportion for different values of the angle and length of the radius. Can they find any connections? (N.B. Students require the tangent ratio to work out the height of the triangle.)
This line of inquiry uses the revised diagram and continues the focus on areas. What is the angle when the area of the sector is half the area of the triangle? (See a solution below.) What about two-thirds of the area of the triangle? Or three-quarters? Is there a pattern from which to generalise?
The inquiry returns to the original diagram (see picture below) with this question from a student: What are the sizes of the two angles if the areas of the three regions are equal?
The question can be answered in parts. Angle α comes from the previous line of inquiry when the area of the triangle is twice the area of the sector. As the two sectors are congruent, θ = 2α.
Students might decide to devise a proof immediately after finding more examples or practising a procedure. The slides contain two versions of a proof - one using degrees (shown below) and the other using radians.
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