Theory question 103
Mensuration of Plane ShapesThe diagram shows the cross-section of a railway tunnel. If |AB| = 100 m and the radius of the arc is 56 m, calculate, correct to the nearest metre, the perimeter of the cross-section.
Show the model answer
Model answer
Let O be the centre. Half of AB = 50 m, so sin ½∠AOB = 50/56 and ½∠AOB ≈ 63.23°, giving ∠AOB ≈ 126.5°.
The arc of the tunnel is the major arc: 360° − 126.5° = 233.5°.
Arc length = 233.5/360 × 2 × 22/7 × 56 ≈ 228.4 m.
Perimeter = 228.4 + 100 ≈ 328 m.
Explanation
The tunnel's curved part is the major arc, so use the reflex angle at the centre.