krit.club logo

Geometry - Understanding Elementary Shapes: Triangles, Quadrilaterals, Polygons

Grade 6ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Triangles are classified by their sides into Equilateral (all sides equal), Isosceles (two sides equal), and Scalene (no sides equal). They are also classified by angles into Acute (all angles < 90∘90^{\circ}), Right (one angle = 90∘90^{\circ}), and Obtuse (one angle > 90∘90^{\circ}).

Diagram showing an equilateral triangle and a right-angled triangle.
•

Quadrilaterals are four-sided polygons. Key types include the Parallelogram (opposite sides parallel and equal), Rectangle (parallelogram with four right angles), Square (rectangle with all sides equal), and Rhombus (parallelogram with all sides equal but not necessarily right angles).

A rectangle and a parallelogram side by side.
•

Polygons are closed plane figures made of line segments. A 'Regular Polygon' has all sides and all angles equal. Examples include a pentagon (55 sides), hexagon (66 sides), and octagon (88 sides).

A regular hexagon showing six equal sides.
•

The diagonal of a polygon is a line segment connecting two non-consecutive vertices. In a quadrilateral, there are exactly two diagonals.

📐Formulae

Sum of interior angles of a triangle = 180∘180^{\circ}

Sum of interior angles of a quadrilateral = 360∘360^{\circ}

Sum of interior angles of a polygon with nn sides = (n−2)×180∘(n - 2) \times 180^{\circ}

Each interior angle of a regular polygon = (n−2)×180∘n\frac{(n - 2) \times 180^{\circ}}{n}

Perimeter of a regular polygon = n×sn \times s (where ss is the length of one side)

💡Examples

Problem 1:

In a triangle ABCABC, the measure of ∠A=55∘\angle A = 55^{\circ} and ∠B=65∘\angle B = 65^{\circ}. Find the measure of the third angle ∠C\angle C.

Solution:

  1. We know that the sum of the angles in a triangle is 180∘180^{\circ}.
  2. Therefore, ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^{\circ}.
  3. Substitute the given values: 55∘+65∘+∠C=180∘55^{\circ} + 65^{\circ} + \angle C = 180^{\circ}.
  4. Add the known angles: 120∘+∠C=180∘120^{\circ} + \angle C = 180^{\circ}.
  5. Subtract 120∘120^{\circ} from both sides: ∠C=180∘−120∘=60∘\angle C = 180^{\circ} - 120^{\circ} = 60^{\circ}.

Explanation:

This problem uses the Angle Sum Property of Triangles to find an unknown interior angle.

Problem 2:

Calculate the sum of the interior angles of a regular Hexagon.

Solution:

  1. A hexagon has n=6n = 6 sides.
  2. The formula for the sum of interior angles is (n−2)×180∘(n - 2) \times 180^{\circ}.
  3. Substitute n=6n = 6 into the formula: (6−2)×180∘(6 - 2) \times 180^{\circ}.
  4. Calculate the subtraction: 4×180∘4 \times 180^{\circ}.
  5. Multiply to find the total: 720∘720^{\circ}.

Explanation:

The sum of interior angles for any polygon depends on the number of triangles it can be divided into, which is (n−2)(n-2).

Problem 3:

Find the measure of each interior angle of a regular Pentagon.

A regular pentagon with an interior angle labeled 108 degrees.

Solution:

  1. A pentagon has n=5n = 5 sides.
  2. The sum of interior angles is given by (n−2)×180∘(n - 2) \times 180^{\circ}.
  3. Sum =(5−2)×180∘=3×180∘=540∘= (5 - 2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ}.
  4. For a regular pentagon, each angle =540∘5=108∘= \frac{540^{\circ}}{5} = 108^{\circ}.

Explanation:

We use the polygon angle sum formula and divide by the number of vertices since all angles in a regular polygon are equal.

Problem 4:

In the given quadrilateral ABCDABCD, three angles are 70∘70^{\circ}, 110∘110^{\circ}, and 80∘80^{\circ}. Find the value of the fourth angle xx.

Quadrilateral ABCD with three known angles and one unknown angle x.

Solution:

  1. Sum of interior angles of a quadrilateral =360∘= 360^{\circ}.
  2. Let the angles be 70∘+110∘+80∘+x=360∘70^{\circ} + 110^{\circ} + 80^{\circ} + x = 360^{\circ}.
  3. 260∘+x=360∘260^{\circ} + x = 360^{\circ}.
  4. x=360∘−260∘=100∘x = 360^{\circ} - 260^{\circ} = 100^{\circ}.

Explanation:

The sum of all four interior angles in any quadrilateral must equal 360 degrees.