Quadrilaterals and polygons for CDS

Polygon questions in CDS are among the quickest in the geometry section. There are three formulas for angles and diagonals, and a table of quadrilateral properties. Most questions test whether you can run a formula backwards: given the interior angle, find the number of sides; given the diagonals, find the side.

Quadrilateral questions add one more layer. You have to know which properties belong to which shape. A rhombus has perpendicular diagonals and a rectangle does not. A rectangle has equal diagonals and a rhombus does not. Mix these up and a two-line question goes wrong.

Polygon formulas, and why they work

For a polygon with n sides:

• Sum of interior angles = (n − 2) × 180°. Why: from one vertex you can draw diagonals that split the polygon into n − 2 triangles.
• Sum of exterior angles = 360°, for any convex polygon. Walk around the boundary and you turn through one full circle.
• For a regular polygon, each exterior angle = 360°/n and each interior angle = 180° − 360°/n.
• Number of diagonals = n(n − 3)/2. Why: each vertex joins n − 3 others by a diagonal (not itself, not its two neighbours), and each diagonal is counted twice.

The exterior-angle route is usually fastest. If a regular polygon has interior angles of 150°, each exterior angle is 30°, so n = 360 ÷ 30 = 12.

Regular polygon table

Quadrilaterals

The angles of any quadrilateral add to 360°. Each special quadrilateral adds its own properties.

A square is both a rectangle and a rhombus, so it has every property of both.

Area formulas

Useful results

• Rhombus: side² = (d₁/2)² + (d₂/2)², because the diagonals cross at right angles and bisect each other.
• Parallelogram: d₁² + d₂² = 2(a² + b²). The sum of the squares of the diagonals equals the sum of the squares of all four sides.
• Joining the midpoints of the sides of any quadrilateral gives a parallelogram with half the original area. For a rectangle this parallelogram is a rhombus, and for a rhombus it is a rectangle.
• The bisectors of the four angles of a parallelogram enclose a rectangle.

Worked questions

Question 1: Each interior angle of a regular polygon is four times its exterior angle. Find the number of sides.

• Interior + exterior = 180°, so 5 × exterior = 180° and the exterior angle is 36°.
• n = 360 ÷ 36 = 10.

Question 2: A polygon has 54 diagonals. How many sides does it have?

• n(n − 3)/2 = 54, so n(n − 3) = 108.
• 12 × 9 = 108, so n = 12.

Question 3: Two adjacent angles of a parallelogram are (2x + 15)° and (3x − 5)°. Find them.

• Adjacent angles add to 180°: 5x + 10 = 180, so x = 34.
• The angles are 83° and 97°.

Question 4: A rhombus has side 13 cm and one diagonal 24 cm. Find the other diagonal and the area.

• Half of the known diagonal is 12. The other half-diagonal = √(169 − 144) = 5, so the diagonal is 10 cm.
• Area = ½ × 24 × 10 = 120 cm².

Question 5: A parallelogram has sides 5 cm and 7 cm and one diagonal of 8 cm. Find the other diagonal.

• d₂² = 2(25 + 49) − 64 = 148 − 64 = 84.
• d₂ = √84 = 2√21 cm, about 9.2 cm.

Question 6: An isosceles trapezium has parallel sides 10 cm and 22 cm, and each slant side is 10 cm. Find its area.

• Figure: drop perpendiculars from the ends of the short side onto the long side. They cut off (22 − 10)/2 = 6 cm on each side.
• Height = √(100 − 36) = 8 cm.
• Area = ½ × (10 + 22) × 8 = 128 cm².

Practice set

1. Find the sum of the interior angles of a pentagon.
2. Each interior angle of a regular polygon is 140°. How many sides does it have?
3. How many diagonals does a decagon have?
4. Each exterior angle of a regular polygon is 30°. Find the sum of its interior angles.
5. The angles of a quadrilateral are in the ratio 3 : 5 : 7 : 9. Find the largest angle.
6. The diagonals of a rhombus are 10 cm and 24 cm. Find its perimeter.
7. A square has a diagonal of 8√2 cm. Find its area.
8. The midpoints of the sides of a rectangle of area 48 cm² are joined. Name the figure formed and find its area.

Answers

1. 540°. (5 − 2) × 180°.
2. 9. The exterior angle is 40°, and 360 ÷ 40 = 9.
3. 35. 10 × 7 ÷ 2.
4. 1,800°. n = 12, and 10 × 180°.
5. 135°. 24 parts make 360°, so one part is 15° and the largest is 9 × 15°.
6. 52 cm. The half-diagonals are 5 and 12, so each side is 13.
7. 64 cm². The side is 8 cm.
8. A rhombus of area 24 cm². The midpoint figure has half the area.

What to do next

• Write the regular polygon table from memory, including the diagonal counts
• Fill in the quadrilateral property table blank, then check it against this one
• Revise lines, angles and triangles, since every polygon result splits into triangles
• Continue with area and perimeter for more area questions on these shapes
• Solve the polygon and quadrilateral questions from the last five CDS papers