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NCERT Solutions for Class 7th Maths Chapter 6 Arch Designs — In-text Questions

Book page 149 Updated on2026-09-19

Q1.
How did they make these arches?
Answer

The stonework starts as a drawing. The mason first sets the arch out full size on a flat surface — paper, board or the stone itself — using nothing but a straight edge and a compass (or a peg and cord on a large job). The stones are then cut to follow those drawn arcs.

  1. Draw the supporting lines that fix the width and the springing points of the arch.
  2. Find the centres of the arcs on those supporting lines.
  3. Sweep the arcs with a compass or a cord.
  4. Cut the stones to the drawn curves and build.
Why it happens: an arch has to be symmetric or it will look wrong and stand badly. Symmetry cannot be judged by eye across several metres of stone. But it can be constructed — equal lengths and equal angles on the two sides — and that is exactly what a compass guarantees.
Did you know? The pictures in the book are of the Diwan-i-Aam at the Red Fort and an arch in Central Park, New York City — the same geometry, centuries and continents apart.
Q2.
Construct this arch shape on a piece of paper.
A trefoil arch made of three arcsABCD
The trefoil arch of page 149, with its four corner points A, B, C and D marked.
Answer

Draw the four support points first — A and D on the base line, B and C above them — and then hang the arcs on that frame.

  1. Draw the base segment AD.
  2. Construct equal angles at A and at D, opening inwards (copy one angle to the other end). This makes ∠BAD = ∠CDA.
  3. With one compass opening, mark B on the arm from A and C on the arm from D, so AB = CD.
  4. Join B to C. The frame ABCD is now symmetric about the perpendicular bisector of AD.
  5. Draw the three arcs of the trefoil on this frame — one small arc over AB, one small arc over CD, and the large arc over BC — adjusting the radii until the curve looks right.
Why it happens: AB = CD and ∠BAD = ∠CDA make the left half of the frame a mirror image of the right half. Once the frame is symmetric, arcs drawn with matching radii on the two sides are symmetric too, and the finished arch balances.
Q3.
For symmetry, we should have AB = CD, and ∠BAD = ∠CDA. How would you construct these support lines?
Answer

One length carried by the compass, one angle carried by the copy-an-angle construction.

What must matchToolHow
∠BAD = ∠CDACompassDraw the angle at A, then copy it at D using arc-and-chord (SSS)
AB = CDCompassSet the compass to AB and cut the arm at D to get C
Copying the angle: arc of radius r from A cuts AD at P and AB at Q
Same radius r from D cuts DA at P′
Transfer the chord PQ from P′ to get Q′
⇒ ΔAPQ ≅ ΔDP′Q′ (SSS) ⇒ ∠BAD = ∠CDA
Why it happens: symmetry about the perpendicular bisector of AD means the left half must fold exactly onto the right half. Folding carries A to D, so it must carry B to C. For that, the angle at A must equal the angle at D and the arm lengths must be equal — precisely the two conditions given.
Tip: construct the perpendicular bisector of AD lightly first. It is the fold line, and it tells you at a glance whether your frame is really symmetric.
Q4.
Use these support lines to construct an arch. If required, adjust the radii of the arcs to make the arch look more aesthetically pleasing.
Answer

With the frame ABCD ready, draw three arcs — but always change the two side arcs together.

  1. Left lobe: choose a centre on the perpendicular bisector of AB and draw an arc from A to B.
  2. Right lobe: use the same compass opening on the perpendicular bisector of CD, and draw the arc from D to C.
  3. Top lobe: choose a centre on the perpendicular bisector of BC (which is also the axis of the whole figure) and draw the arc from B to C.
  4. Step back and look. A smaller radius makes the lobes rounder and the arch chubbier; a larger radius flattens them and the arch looks taller and calmer. Redraw until it pleases you.
Why it happens: the frame fixes where the arcs begin and end; the radius decides only how much they bulge. So you can change the mood of the arch freely without ever losing its symmetry — as long as the left and right radii stay equal.
Try This: draw the same frame three times and use three different radii. Put the three arches side by side and decide which one you would build.
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