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Properties of 2D shapes

Properties of 2D shapes

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Tutor: Bilal

Summary

Properties of 2D shapes

In a nutshell

2D shapes such as triangles and quadrilaterals can be further classified into more distinct shapes that each have their own properties. There are also two different types of symmetry that can be applied to 2D shapes - line symmetry and rotational symmetry. Regular polygons can also be classified by their internal and external angles.



Symmetry

There are two different types of symmetry: line symmetry and rotational symmetry.


Line symmetry

Line symmetry of a shape is when you can draw a mirror line across the shape. A mirror line is a line that can be drawn that acts like a mirror; it reflects one side of the line onto the other.


Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes
Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes


Rotational symmetry

Rotational symmetry is when you can rotate the shape and it looks exactly the same.

The order of rotational symmetry is a property that tells you how many times the shape looks the same when you rotate it a full circle.



This shape has rotational symmetry of order 22​ because rotating it 180180^\circ​ will make it the same shape again.



Regular polygons

A regular polygon is a 2D shape that has the same side lengths and angles.


NUMBER OF SIDES

NAME OF REGULAR POLYGON

SHAPE

33​​

Equilateral triangle

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

44​​

Square

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

55​​

Regular pentagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

66​​

Regular hexagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

77​​

Regular heptagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

88​​

Regular octagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

99​​

Regular nonagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

1010​​

Regular decagon

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes



Triangles

There are four main types of triangles.


NAME

DESCRIPTION

LINES OF SYMMETRY

ORDER OF ROTATIONAL SYMMETRY

SHAPE

Equilateral triangle

All sides of equal length, all angles are 6060^\circ

33​​

33​​


Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Right-angled triangle

11​ angle of 9090^\circ

00, or 11 if it's a right-angled isoceles triangle

00​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Isosceles triangle

22​ equal sides and 22​ equal angles

11​​

00​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Scalene triangle

No equal sides or angles

00​​

00​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes



​​Quadrilaterals

A quadrilateral is the name given to a shape that has four sides. There are six main types of quadrilaterals.


NAME

DESCRIPTION

LINES OF SYMMETRY

ORDER OF ROTATIONAL SYMMETRY

SHAPE

Square

44​ right angles, 44​ sides of equal length

44​​

44​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Rectangle

44​ right-angles

22​​

22​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Rhombus

44​ sides of equal length, 22 pairs of equal angles

22​​

22​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Parallelogram

22​ pairs of parallel equal sides, 22​ pairs of equal angles

00​​

22​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Trapezium

11​ pair of parallel sides

00, or 11 in the case of an isoceles trapezium

00​​
Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes

Kite

22​ pairs of equal sides, 11​ pair of equal angles

11​​

00​​

Maths; Shapes and area; KS4 Year 10; Properties of 2D shapes



Interior and exterior angles

An interior angle of a polygon is the inner angle formed by two sides of the shape.

An exterior angle of a polygon is the angle formed by extending one side of the shape past the vertex.


Finding the interior angle of a regular polygon

The sum of the interior angles of a polygon is found by using the formula:

θ=(n2)×180\theta =(n-2)\times180​​


Where θ\theta is the sum of the interior angles, and nn​ is the number of sides.


Example 1

What is the size of an interior angle of a regular nonagon?


A nonagon has 99 sides, so n=9n=9. The sum of the interior angles is therefore given to be:

θ=(92)×180=7×180\theta=(9-2)\times180=7\times180​​

θ=1260\theta=1260^\circ​​


To find the size of one interior angle, divide this number by 99:

1260÷9=1401260\div9=140​​


The size of one interior angle in a nonagon is 140\underline{140^\circ}.


Finding the exterior angle of a regular polygon

Exterior angles of a regular polygon always add up to 360360^\circ. Therefore, the formula for one exterior angle is given to be:

θ=360n\theta=\frac{360}{n}​​


Where θ\theta is the size of one exterior angle, and nn​ is the number of sides.


Example 2

What is the size of one exterior angle of a nonagon?


As in the example above, n=9n=9. Hence:

θ=3609=40\theta=\frac{360}{9}=40


The size of one exterior angle of a nonagon is 140\underline{140^\circ}.


Note: Notice how the exterior and interior angle of a regular polygon add up to 180180^\circ. This is always true because both angles are on the same straight line, and the angles in a straight line add up to 180180^\circ.



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FAQs - Frequently Asked Questions

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