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 Classified by Sides: Triangles are 3-sided polygons. An Equilateral triangle has all three sides equal and all angles measure 60∘60^{\circ}. An Isosceles triangle has at least two equal sides and two equal base angles. A Scalene triangle has no equal sides and no equal angles, appearing as an asymmetrical shape.

•

Triangles Classified by Angles: An Acute-angled triangle has all angles less than 90∘90^{\circ}. A Right-angled triangle has exactly one angle equal to 90∘90^{\circ}, forming an 'L' shape at one vertex. An Obtuse-angled triangle has one angle greater than 90∘90^{\circ} but less than 180∘180^{\circ}, making it look wide or spread out.

•

Parallelograms and Rectangles: A Parallelogram is a quadrilateral where opposite sides are parallel and equal in length. A Rectangle is a special type of parallelogram where every interior angle is exactly 90∘90^{\circ}, forming a perfectly square-cornered four-sided shape.

•

Squares and Rhombuses: A Rhombus is a parallelogram with all four sides of equal length, often looking like a tilted diamond. A Square is a regular quadrilateral that combines properties of both a rectangle and a rhombus, having four equal sides and four 90∘90^{\circ} angles.

•

Trapeziums and Kites: A Trapezium (or Trapezoid) is a quadrilateral with at least one pair of parallel sides; it often looks like a triangle with the top point cut off parallel to the base. A Kite has two pairs of equal-length sides that are adjacent to each other, resembling the classic flying toy shape.

•

Polygon Classification: Polygons are closed figures made of line segments. They are named by their number of sides: Pentagon (5 sides), Hexagon (6 sides), Heptagon (7 sides), and Octagon (8 sides). A 'Regular Polygon' has all sides and all interior angles equal.

•

Convex and Concave Polygons: In a Convex polygon, all interior angles are less than 180∘180^{\circ} and all vertices point outwards. In a Concave polygon, at least one interior angle is 'reflex' (greater than 180∘180^{\circ}), creating an indentation or 'cave' in the shape's boundary.

📐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).