# Shapes

## Areas of 2D shapes

These are some common two-dimensional shapes and the formulae for calculating their areas:

### Triangle

The area of a triangle is half its height times its base:

\$A = ½h⋅b\$

This works for all triangles. If you made a copy of a triangle, and cut up the copy so that you made a rectangle with the original triangle, you would see that the rectangle has the dimensions of the height and base of the triangle. The area is therefore half of this rectangle!

### Heron's formula

The area of a triangle can also be calculated using Heron's formula:

\$\$A = √{s(s-a)(s-b)(s-c)}\$\$

Where \$a\$, \$b\$ and \$c\$ are le lengths of the sides of the triangle, and \$s = {a+b+c}/2\$

Square: \$A = L^2\$, where L is the length of one side.

Rectangle: \$A = L ⋅ W\$, where L is the length, and W is the width.

Rhomboid (elongated diamond): \${pq}/2\$, where p and q are the lengths of the two diagonals.

Rhombus ('pushed over rectangle'): \$L ⋅ H\$, where L is the length and H is the height (perpendicular to L).

## Triangles

In two dimensions (such as on a flat piece of paper), the angles of a triangle all add up to 180°.

The equilateral triangle has all three side and all three angles equal. The angles are 60° each.

The isoceles triangle has two sides the same length. This requires two angles to be same as well.

Trapezium: one set of two parallel sides
Isoceles trapezium: two sides equal length, the other two sides parallel
Parallelogram: two sets of parallel and equal length sides
Rhombus: two sets of two parallel sides, 4 equal lengths, two sets of identical angles

## Lines of Symmetry

A useful way to describe a shape is to state its number of reflection lines of symmetry. These are the number of straight lines that may be drawn through a shape, across which a reflection of the shape would result in an identical shape.

### Triangle lines of symmetry

Triangles may have zero, one or three lines of symmetry, depending on their type.

An equilateral triangle has 3 lines of symmetry.

Equilateral triangles have 3 lines of symmetry
Line of symmetry for an isoceles triangle

An isoceles has only one.

Other types of triangles have no lines of symmetry, so their orientation is unique.

A square reflects across a horizontal line through its centre, and a vertical line through its centre. It also has lines of symmetry across its diagonals. It therefore has four lines of reflection symmetry.

A rhombus, like a rectangle, has 2 lines of symmetry

A rhombus and a rectangle have only 2 lines of symmetry.

A parallelogram has no lines of symmetry.

## Rotational Symmetry

A shape which has rotational or radial symmetry is one which, when rotated 360° reassumes an identical form to the starting position one or more times.

Order of rotational symmetry: the number of times an object takes an identical shape while being rotated through 360°. e.g. a square has rotational symmetry of 4, a rhombus 2, a triangle 3.

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