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:

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

Quadrilaterals

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.

Quadrilaterals

Quadrilaterals are four-sided geometric shapes.

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.

Quadrilateral lines of symmetry

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.