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CBSE · Class 8 · Mathematics Part 1

Part 1, Chapter 4: Quadrilaterals

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Angle sum and basics of quadrilaterals

From daily life

Take a rectangular photo frame and push one corner. The shape changes, but if three corners are known, the fourth corner is already fixed: the four angles always make one full turn, 360°.

Simple definition

A quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals. The sum of its four interior angles is 360°, because a diagonal splits it into two triangles (2 × 180°). The exterior angles (one at each vertex) also add up to 360°.

180°180°ABCD

In the figure: Two triangles, each 180°, together fill all four angles of the quadrilateral.

Rule 1: Find the missing angle — Add the three known angles and subtract from 360°.

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Angle sum and basics of quadrilaterals

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  • Easy: 85° + 95° + 110° = 290°, so the fourth angle is 360° − 290° = 70°.
  • Medium: 100° + 80° + 75° = 255°, so the fourth angle is 105°.
  • Exam-type: Angles x, x + 20°, x + 40°, x + 60°: 4x + 120° = 360°, so x = 60° (angles 60°, 80°, 100°, 120°).

Rule 2: Angles in a ratio — Add the ratio parts, find the angle for one part (360° ÷ total) and multiply.

  • Easy: Ratio 1 : 2 : 3 : 4: 360° ÷ 10 = 36°, so the largest angle is 4 × 36° = 144°.
  • Medium: Ratio 2 : 3 : 3 : 4: 360° ÷ 12 = 30°, so the smallest angle is 2 × 30° = 60°.
  • Exam-type: Ratio 3 : 4 : 5 : 6: 360° ÷ 18 = 20°, so the angles are 60°, 80°, 100°, 120°.

Rule 3: Interior and exterior angles — Interior angle + exterior angle at a vertex = 180° (a straight line).

  • Easy: Interior 115° gives exterior 180° − 115° = 65°.
  • Medium: Three exterior angles 80°, 100°, 70°: the fourth is 360° − 250° = 110°.
  • Exam-type: One angle 105° and the other three equal: each is (360° − 105°) ÷ 3 = 85°.
Remember

If one angle is more than 180° (a reflex angle), the quadrilateral is concave and one corner points inwards. The angle sum is still 360°.

Common mistake

Using 180° (the triangle sum) for a quadrilateral. Correct: 360°.

Shortcut

Triangle 180°; each extra side adds one more triangle, so quadrilateral = 2 triangles = 360°. Memory clue: "Triangle 180, Four-sided 360".

Spot the difference

PointTriangleQuadrilateral
Sides34
Angle sum180°360°
Diagonals02

Practice questions

Easy
  1. 1Three angles of a quadrilateral are 90°, 110°, 70°. Find the fourth.
  2. 2Angles are in the ratio 1 : 1 : 2 : 2. Find the largest angle.
  3. 3An interior angle is 125°. Find the exterior angle.
Medium
  1. 4The angles are x, x + 30°, x + 60°, x + 90°. Find the greatest angle.
  2. 5Three exterior angles are 80°, 100°, 70°. Find the fourth exterior angle.
  3. 6Three angles are 95°, 85°, 100°. Find the fourth.
Challenging
  1. 7Angles are in the ratio 4 : 5 : 6 : 9. Which angle is a right angle? State its value.
  2. 8Two angles are 80° and 100°. The other two are equal. Find each.
Show answers (try them yourself first)
  1. 190°
  2. 2120°
  3. 355°
  4. 4135°
  5. 5110°
  6. 680°
  7. 7The 6-part angle: 6 × 15° = 90°
  8. 890° each

Rectangle and square

From daily life

A notebook page, a window pane and a carrom board are rectangles; a chess board, a floor tile and a postage stamp are usually squares. Corners are right angles and opposite edges match.

Simple definition

A rectangle has four right angles, equal opposite sides, and equal diagonals that bisect each other. A square is a rectangle with all sides equal; its diagonals are also perpendicular. Perimeter = 2 × (l + w), area = l × w.

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Rectangle and square

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Rule 1: Perimeter and area — Perimeter is the boundary length; area is the space inside.

  • Easy: 12 cm × 5 cm: perimeter 2 × 17 = 34 cm, area 60 cm².
  • Medium: Perimeter 46 cm and length 15 cm: half perimeter 23, so width = 23 − 15 = 8 cm.
  • Exam-type: A 30 m × 20 m park fenced at ₹15 per metre: 100 × 15 = ₹1500.

Rule 2: Diagonals of a rectangle — Both diagonals are equal and bisect each other. A diagonal makes a right triangle: d² = l² + w².

  • Easy: One diagonal 18 cm, so the other is also 18 cm.
  • Medium: 15 cm × 8 cm: d = √(225 + 64) = √289 = 17 cm.
  • Exam-type: ∠OAB = 35° and OA = OB, so ∠AOB = 180° − 35° − 35° = 110°.

Rule 3: Square — All sides equal: perimeter = 4 × side, area = side × side. Diagonals are equal and perpendicular.

  • Easy: Perimeter 36 cm: side 9 cm, area 81 cm².
  • Medium: Half of the diagonal is 7 cm, so the diagonal is 14 cm.
  • Exam-type: Perimeter 48 cm: side 12 cm, area 144 cm².
Remember

A rectangle has 2 lines of symmetry and a square has 4. Every square is a rectangle, but not every rectangle is a square.

Common mistake

Thinking the diagonals of a rectangle are perpendicular. They are perpendicular only when the rectangle is a square.

Shortcut

For a rectangle diagonal use the famous triples 3-4-5, 5-12-13, 8-15-17, 9-40-41. Memory clue: "Three-four-five, five-twelve-thirteen".

Spot the difference

PointRectangleSquare
Sidesopposite sides equalall four equal
Diagonalsequal, bisect each otherequal, perpendicular, bisect
Lines of symmetry24

Practice questions

Easy
  1. 1A rectangle is 9 cm × 7 cm. Find its area.
  2. 2A square has side 8 cm. Find its perimeter.
  3. 3One diagonal of a rectangle is 20 cm. What is the other?
Medium
  1. 4Find the diagonal of a rectangle 20 cm × 15 cm.
  2. 5The perimeter of a rectangle is 40 cm and its length is 12 cm. Find the width.
  3. 6A square has area 49 cm². Find its perimeter.
Challenging
  1. 7Find the diagonal of a rectangle 9 cm × 40 cm.
  2. 8A 40 m × 25 m field is fenced at ₹20 per metre. Find the cost.
Show answers (try them yourself first)
  1. 163 cm²
  2. 232 cm
  3. 320 cm
  4. 425 cm
  5. 58 cm
  6. 628 cm
  7. 741 cm
  8. 82 × 65 = 130 m; 130 × 20 = ₹2600

Parallelogram

From daily life

A sliding folding gate (or a pantograph) keeps both pairs of opposite bars parallel even when the gate opens or closes. Its shape is a parallelogram: the angles change but the opposite sides stay equal and parallel.

Simple definition

A parallelogram has both pairs of opposite sides parallel. Then: opposite sides are equal, opposite angles are equal, adjacent angles add up to 180°, the diagonals bisect each other, and a diagonal makes two congruent triangles. Area = base × height.

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Parallelogram

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Rule 1: Angles — Opposite angles are equal; adjacent angles are supplementary (add up to 180°).

  • Easy: One angle 65°: the opposite is 65° and each adjacent angle is 115°.
  • Medium: Adjacent angles in ratio 2 : 3: 180° ÷ 5 = 36°, so 72° and 108°.
  • Exam-type: ∠A = 2x + 10° and ∠C = 3x − 20° (opposite): x = 30°, so ∠A = 70°.

Rule 2: Sides and diagonals — Opposite sides are equal. The diagonals bisect each other (but need not be equal).

  • Easy: AB = 7 cm, BC = 5 cm: perimeter = 2 × 12 = 24 cm.
  • Medium: AO = 6 cm, so the whole diagonal AC = 12 cm.
  • Exam-type: Opposite sides x + 3 and 2x − 4: x = 7, so each is 10.

Rule 3: Area — Area = base × height, where the height is the straight (perpendicular) distance between the two parallel sides.

  • Easy: Base 14 cm, height 9 cm: 14 × 9 = 126 cm².
  • Medium: Area 90 cm², base 15 cm: height = 90 ÷ 15 = 6 cm.
  • Exam-type: A field with base 40 m and height 25 m, levelled at ₹12 per m²: 1000 × 12 = ₹12,000.
Remember

Rectangle, rhombus and square are special parallelograms. A parallelogram with one right angle is a rectangle.

Common mistake

Using the slanting side as the height. The height must be perpendicular to the base.

Shortcut

Area uses the straight height, never the slant side. Memory clue: "Height stands straight".

Spot the difference

PointParallelogramRectangle
Anglesopposite equal, adjacent add to 180°90°
Diagonalsbisect each otherequal and bisect each other
Areabase × heightl × w

Practice questions

Easy
  1. 1One angle of a parallelogram is 80°. What is the opposite angle?
  2. 2Find the adjacent angle of 80°.
  3. 3Two sides are 6 cm and 4 cm. Find the perimeter.
Medium
  1. 4Base 12 cm, height 5 cm. Find the area.
  2. 5The diagonals meet at O and BO = 4.5 cm. Find BD.
  3. 6Adjacent angles are in the ratio 4 : 5. Find the smaller angle.
Challenging
  1. 7∠B = 3x + 5° and ∠D = 5x − 25°. Find ∠B.
  2. 8The perimeter is 56 cm and the adjacent sides are in the ratio 3 : 4. Find both sides.
Show answers (try them yourself first)
  1. 180°
  2. 2100°
  3. 320 cm
  4. 460 cm²
  5. 59 cm
  6. 680°
  7. 750°
  8. 812 cm and 16 cm (28 ÷ 7 = 4)

Rhombus and kite

From daily life

A diamond-shaped floor tile is a rhombus, and a paper kite flying in the sky is a kite shape. Both have two diagonals that cross at right angles.

Simple definition

A rhombus has all four sides equal; it is a parallelogram whose diagonals are perpendicular bisectors and bisect its angles. A kite has two pairs of equal adjacent sides and perpendicular diagonals. For both, area = ½ × (product of the diagonals).

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Rhombus and kite

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Rule 1: Side and perimeter of a rhombus — Perimeter = 4 × side. If the diagonals are given, half-diagonals form a right triangle with the side.

  • Easy: Side 9 cm: perimeter 36 cm.
  • Medium: Perimeter 52 cm: side 52 ÷ 4 = 13 cm.
  • Exam-type: Diagonals 10 cm and 24 cm: side = √(5² + 12²) = 13 cm.

Rule 2: Area and angles of a rhombus — Area = ½ × d₁ × d₂. Diagonals bisect the angles.

  • Easy: Diagonals 16 and 12: area = ½ × 16 × 12 = 96 cm².
  • Medium: Area 84 cm², d₁ = 12 cm: ½ × 12 × d₂ = 84, so d₂ = 14 cm.
  • Exam-type: ∠A = 80°: diagonal AC splits it into 40° and 40°; the adjacent angle is 100°.

Rule 3: Kite — Perimeter = 2 × (a + b). Area = ½ × d₁ × d₂. The angles between the unequal sides are equal.

aabb

In the figure: Kite: a, a and b, b are the equal adjacent sides; the diagonals cross at 90°.

  • Easy: Sides 5, 5, 8, 8: perimeter = 26 cm.
  • Medium: Diagonals 10 and 18: area = ½ × 10 × 18 = 90 cm².
  • Exam-type: ∠A = 70°, ∠C = 110°, ∠B = ∠D: 2∠B = 180°, so ∠B = ∠D = 90°.
Remember

A rhombus has 2 lines of symmetry (its diagonals); a general kite has 1. A square is both a rhombus and a rectangle.

Common mistake

Saying the diagonals of a rhombus are equal (that is a square), or using area = side × side for a rhombus.

Shortcut

Half the diagonals and the side make a right triangle: side² = (d₁/2)² + (d₂/2)². Memory clue: "Half, half, then Pythagoras".

Spot the difference

PointRhombusKite
Equal sidesall fourtwo pairs of adjacent sides
Opposite sides parallel?yesno (in general)
Diagonalsperpendicular, bisect each otherperpendicular, only one is bisected
Lines of symmetry21

Practice questions

Easy
  1. 1A rhombus has side 7 cm. Find its perimeter.
  2. 2The diagonals of a rhombus are 8 cm and 6 cm. Find its area.
  3. 3A kite has sides 4, 4, 9, 9 cm. Find its perimeter.
Medium
  1. 4The diagonals of a rhombus are 6 cm and 8 cm. Find its side.
  2. 5One angle of a rhombus is 110°. Find the adjacent angle.
  3. 6The diagonals of a kite are 12 cm and 7 cm. Find its area.
Challenging
  1. 7A rhombus has side 10 cm and one diagonal 12 cm. Find the other diagonal.
  2. 8In kite ABCD, ∠A = 100° and ∠B = ∠D = 60°. Find ∠C.
Show answers (try them yourself first)
  1. 128 cm
  2. 224 cm²
  3. 326 cm
  4. 45 cm
  5. 570°
  6. 642 cm²
  7. 716 cm (half = 8, since 6² + 8² = 10²)
  8. 8140°

Trapezium and classification of quadrilaterals

From daily life

The side view of a water tumbler or a bucket is a trapezium: the top and the bottom are parallel, but the slanting sides are not. Many road signs and bridge girders also use this shape.

Simple definition

A trapezium has exactly one pair of parallel sides (the bases). In an isosceles trapezium the slanting sides are equal, base angles are equal and the diagonals are equal. Area = ½ × (sum of parallel sides) × height. Angles on the same slanting side add up to 180°.

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Trapezium and classification of quadrilaterals

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Rule 1: Angles of a trapezium — If AB ∥ CD, the angles on the same slanting side (∠A and ∠D, ∠B and ∠C) add up to 180°.

  • Easy: ∠A = 75°, so ∠D = 180° − 75° = 105°.
  • Medium: Isosceles, ∠A = 70°: ∠B = 70°, ∠C = ∠D = 110°.
  • Exam-type: Right trapezium with ∠A = 90°: ∠D = 90°, and ∠B + ∠C = 180°.

Rule 2: Area and middle segment — Area = ½ × (a + b) × h. The segment joining midpoints of the slanting sides is parallel to the bases and has length (a + b) ÷ 2.

  • Easy: Bases 10 and 14, height 6: ½ × 24 × 6 = 72 cm².
  • Medium: Area 90 cm², bases 8 and 12: 10h = 90, so h = 9 cm.
  • Exam-type: Bases 9 and 15: the middle segment is (9 + 15) ÷ 2 = 12 cm.

Rule 3: Classify using clues — Ask: Which sides are parallel? How do the diagonals behave? Are there right angles?

  • Easy: Equal diagonals that bisect each other, not perpendicular: rectangle.
  • Medium: Perpendicular diagonals that bisect each other, not equal: rhombus.
  • Exam-type: Equal, perpendicular and bisecting diagonals: square.
Remember

Family tree: a square is a rectangle, a rhombus and a parallelogram; a rectangle or rhombus is a parallelogram; a kite and a trapezium are separate families.

Common mistake

Believing every rhombus is a square, or every trapezium has two pairs of parallel sides. Correct: a square is a special rhombus; a trapezium has just one pair.

Shortcut

Ask three questions in order: parallel sides? diagonals? right angles? Memory clue: "Parallel, Diagonals, Degrees".

Spot the difference

PointTrapeziumParallelogram
Pairs of parallel sides12
Opposite sides equal?not necessarilyalways
Diagonals bisect each other?no (in general)yes

Practice questions

Easy
  1. 1How many pairs of parallel sides does a trapezium have?
  2. 2In trapezium ABCD, AB ∥ CD and ∠A = 60°. Find ∠D.
  3. 3Which quadrilateral has exactly 4 lines of symmetry?
Medium
  1. 4Bases 7 cm and 9 cm, height 5 cm. Find the area.
  2. 5An isosceles trapezium has bases 10 cm and 4 cm and slanting sides 5 cm each. Find the perimeter.
  3. 6Equal diagonals bisect each other but are not perpendicular. Name the quadrilateral.
Challenging
  1. 7The area of a trapezium is 120 cm², its height is 8 cm and one base is 11 cm. Find the other base.
  2. 8Opposite sides are parallel, all sides are equal and no angle is 90°. Name the quadrilateral.
Show answers (try them yourself first)
  1. 11
  2. 2120°
  3. 3Square
  4. 440 cm²
  5. 524 cm
  6. 6Rectangle
  7. 719 cm (11 + b = 30)
  8. 8Rhombus