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Geometry - Properties of 2D shapes (polygons, circles)

Grade 4Cambridge (IGCSE)

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

🔑Concepts

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A polygon is a closed 2D shape with straight sides. Regular polygons have all sides equal and all interior angles equal. For a polygon with nn sides, the sum of interior angles is (n−2)×180∘(n - 2) \times 180^{\circ}.

A regular pentagon with 5 equal sides.
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Quadrilaterals are four-sided polygons. Specific types include the Parallelogram (opposite sides parallel and equal), Rhombus (all sides equal, opposite sides parallel), and Trapezium (one pair of parallel sides).

A parallelogram showing opposite sides are parallel.
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The circle is defined by its center and radius (rr). The distance across the circle through the center is the diameter (dd), where d=2rd = 2r. The boundary length is called the circumference.

Circle with center O and radius r.
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Triangles are classified by sides (Equilateral, Isosceles, Scalene) or by angles (Acute, Right-angled, Obtuse). The sum of angles is always 180∘180^{\circ}.

📐Formulae

Perimeter of a Polygon=Sum of all side lengths\text{Perimeter of a Polygon} = \text{Sum of all side lengths}

Area of a Rectangle=Length×Width\text{Area of a Rectangle} = \text{Length} \times \text{Width}

Diameter=2×Radius\text{Diameter} = 2 \times \text{Radius}

Sum of interior angles in any triangle=180∘\text{Sum of interior angles in any triangle} = 180^{\circ}

💡Examples

Problem 1:

A regular hexagon has one side length of 7 cm. What is its perimeter?

Solution:

42 cm

Explanation:

A hexagon has 6 sides. Since it is a 'regular' hexagon, all 6 sides are equal. Perimeter = 6×7 cm=42 cm6 \times 7\text{ cm} = 42\text{ cm}.

Problem 2:

If the radius of a circle is 5.5 cm, calculate the diameter.

Solution:

11 cm

Explanation:

The diameter is always twice the length of the radius. Diameter = 2×5.5 cm=11 cm2 \times 5.5\text{ cm} = 11\text{ cm}.

Problem 3:

In a right-angled triangle, one angle is 90∘90^{\circ} and another is 45∘45^{\circ}. Find the size of the third angle.

Solution:

45^{\circ}

Explanation:

The sum of angles in a triangle is 180∘180^{\circ}. So, 180∘−(90∘+45∘)=180∘−135∘=45∘180^{\circ} - (90^{\circ} + 45^{\circ}) = 180^{\circ} - 135^{\circ} = 45^{\circ}.

Problem 4:

Identify the quadrilateral that has two pairs of parallel sides, but no right angles and all four sides are equal.

Solution:

Rhombus

Explanation:

A square has four equal sides and right angles. A rhombus has four equal sides but its interior angles are not 90∘90^{\circ}.

Problem 5:

Calculate the perimeter of the following kite, where the short sides are 5 cm5\text{ cm} and the long sides are 9 cm9\text{ cm}.

A kite shape with side labels 5 cm and 9 cm.

Solution:

Perimeter=5+5+9+9=28 cm\text{Perimeter} = 5 + 5 + 9 + 9 = 28\text{ cm}

Explanation:

A kite has two pairs of adjacent sides that are equal in length. To find the perimeter, sum all four side lengths: 2×5 cm+2×9 cm2 \times 5\text{ cm} + 2 \times 9\text{ cm}.

Problem 6:

In the triangle shown, find the value of the missing angle xx.

Triangle with base angles 65 and 40 degrees, and top angle x.

Solution:

x=180∘−(65∘+40∘)x = 180^{\circ} - (65^{\circ} + 40^{\circ}) x=180∘−105∘x = 180^{\circ} - 105^{\circ} x=75∘x = 75^{\circ}

Explanation:

The sum of interior angles in any triangle is 180∘180^{\circ}. Subtract the two known angles from 180∘180^{\circ} to find the unknown third angle.