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Properties of 2D Shapes

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The names and properties of triangles, quadrilaterals and other polygons: equal sides and angles, parallel sides and lines of symmetry, and how to use them to work out lengths and angles.

2D shapes in the exam

A 2D shape is a flat shape, such as a triangle or a square. This page covers the names of shapes and their properties: which sides are equal, which sides are parallel and which angles are equal.

Early in a paper, you are often asked to write down the mathematical name of a shape, for 1 mark. You may be shown the shape, or given its properties in words. You may also be asked to draw a shape, such as a hexagon or a parallelogram.

Later questions, for 2 to 5 marks, tell you the name of a shape and expect you to use its properties to work out a length or an angle. Some of these need an equation.

Learn the names, especially isosceles, parallelogram, kite and trapezium. A spelling mistake is allowed if the name is clear, but a description is not a name.

Marks on a diagram

Diagrams use small marks to show facts. Sides with the same number of dashes across them are equal in length.

A small square in a corner shows a right angle, 90^\circ. Two lines that meet at a right angle are perpendicular.

Arrows on two sides show that they are parallel: they go in the same direction and never meet. Sides with the same number of arrows are parallel to each other.

Diagrams in the exam are often not drawn accurately. Only use a fact that the marks or the words give you. A shape that looks like a square may not be one.

Naming polygons

A polygon is a flat shape whose sides are all straight lines, joined up with no gaps. Its corners are called vertices, and one corner is a vertex.

  • Triangle: 3 sides
  • Quadrilateral: 4 sides
  • Pentagon: 5 sides
  • Hexagon: 6 sides
  • Octagon: 8 sides

A regular polygon has all its sides equal and all its angles equal. A square is a regular quadrilateral, and an equilateral triangle is a regular triangle.

The name depends only on the number of sides. Both shapes in the picture have 6 sides, so both are hexagons, even though the L shape is not regular.

Worked example 1: Naming a polygon

Here is a polygon shaped like an arrow. Write down the mathematical name of this polygon.

  1. Count the sides. Mark each side as you count it, so you don't miss one or count one twice.

  2. Your turn. How many sides does the polygon have?

    The polygon has 6 sides: 1 along the top, 2 round the point, 1 along the bottom and 2 round the notch.

  3. A polygon with 6 sides is a hexagon.
  4. Its sides are not all the same length, so it is not regular, but it is still a hexagon.
  5. Answer: hexagon.

Types of triangle

Triangles are named by their sides and angles. The angles in any triangle add up to 180^\circ

  • Equilateral: all 3 sides equal and all 3 angles equal, each 60^\circ
  • Isosceles: 2 sides equal and 2 angles equal
  • No equal sides: all 3 sides different and all 3 angles different
  • Right-angled: one angle is 90^\circ

In an isosceles triangle, the two equal angles are at the two ends of the side that is different. They are called the base angles.

A triangle can have two names. A triangle with angles of 90^\circ, 45^\circ and 45^\circ is right-angled and isosceles.

Worked example 2: The angles of an isosceles triangle

In triangle DEF, sides DE and DF are equal. Angle DEF is 52^\circ. Write down the mathematical name of triangle DEF. Then work out the size of angle EDF.

  1. Two sides, DE and DF, are equal, so triangle DEF is isosceles.
  2. The equal angles are at E and F, the two ends of the third side, EF. So angle DFE is also 52^\circ
  3. The two base angles add up to 52+52=104^\circ

  4. Your turn. Angles in a triangle add up to 180^\circ. Work out 180-104

    Angle EDF =180-104=76^\circ

  5. Check: 52+52+76=180
  6. Answer: isosceles, and angle EDF =76^\circ

Parallelograms, rectangles, rhombuses and squares

A quadrilateral is any shape with 4 straight sides. The angles in any quadrilateral add up to 360^\circ

A parallelogram has two pairs of parallel sides. Its opposite sides are equal and its opposite angles are equal. Two angles next to each other add up to 180^\circ, such as 70^\circ and 110^\circ

  • Rectangle: four right angles, and opposite sides equal
  • Rhombus: four equal sides, and opposite angles equal
  • Square: four equal sides and four right angles

All four shapes have two pairs of parallel sides, so a rectangle, a rhombus and a square are all special parallelograms. Give the most exact name: a quadrilateral with four equal sides and four right angles is a square.

Worked example 3: Angles in a parallelogram

ABCD is a parallelogram. Angle DAB is 64^\circ. Work out the size of angle ABC.

  1. Opposite angles of a parallelogram are equal, so angle BCD is also 64^\circ
  2. Together, angles DAB and BCD make 64+64=128^\circ

  3. Your turn. The angles in a quadrilateral add up to 360^\circ. Work out 360-128

    Angles ABC and ADC make 360-128=232^\circ together

  4. Angles ABC and ADC are opposite, so they are equal. Each one is 232\div2=116^\circ
  5. Check: angles next to each other add up to 180^\circ, and 64+116=180
  6. Answer: angle ABC =116^\circ

Kites and trapeziums

A trapezium has exactly one pair of parallel sides. The other two sides are not parallel, and they can be any lengths.

A kite has two pairs of equal sides, and the sides in each pair are next to each other, not opposite. It has no parallel sides and one line of symmetry. The two angles where a short side meets a long side are equal.

To name a quadrilateral, count its pairs of parallel sides first. Two pairs: a parallelogram, rectangle, rhombus or square. One pair: a trapezium. No pairs: it may be a kite. Then look at the sides and angles to decide.

Worked example 4: The fourth corner of a kite

P is the point (1, 3), Q is (4, 6) and R is (10, 3). PQRS is a kite, and the line PR is its line of symmetry. Work out the coordinates of S.

  1. PR is a line of symmetry, so S is the reflection of Q in the line PR.
  2. Every point on PR has a y-coordinate of 3. Q has a y-coordinate of 6, so Q is 3 squares above PR.

  3. Your turn. S is 3 squares below PR. What is the y-coordinate of S?

    The y-coordinate of S is 3-3=0

  4. S is directly below Q, so it has the same x-coordinate as Q, which is 4
  5. Check: PQ and PS are equal, and QR and RS are equal, so PQRS is a kite.
  6. Answer: S is (4, 0)

Worked example 5: Parallelogram angles with algebra

PQRS is a parallelogram. Angle SPQ is (2x+16)^\circ and angle QRS is (4x-22)^\circ. Work out the size of angle PQR.

  1. Angles SPQ and QRS are opposite angles of the parallelogram, so they are equal.
  2. 4x-22=2x+16
  3. Subtract 2x from both sides and add 22 to both sides: 2x=38, so x=19
  4. Angle SPQ =2\times19+16=54^\circ. Check: angle QRS =4\times19-22=54^\circ

  5. Your turn. Angles next to each other in a parallelogram add up to 180^\circ. Work out the size of angle PQR.

    Angle PQR =180-54=126^\circ

  6. Answer: angle PQR =126^\circ

Worked example 6: A kite problem with algebra

PQRS is a kite. Sides PQ and PS are each (x+5) cm long. Sides QR and RS are each (3x-4) cm long. The perimeter of the kite is 50 cm. Work out the length of QR.

  1. The perimeter is the total length of all four sides. A kite has two pairs of equal sides, so add each length twice.
  2. (x+5)+(x+5)+(3x-4)+(3x-4)=50
  3. Collect like terms: 8x+2=50, so 8x=48

  4. Your turn. Divide both sides by 8. What is x?

    x=48\div8=6

  5. QR =3\times6-4=14 cm. Check: PQ =6+5=11 cm, and 11+11+14+14=50
  6. Answer: QR =14 cm

Common mistakes

  • Miscounting the sides of a polygon. Mark each side as you count it: a shape with 6 sides is a hexagon, not a pentagon.
  • Calling a trapezium a parallelogram. A trapezium has exactly one pair of parallel sides, and a parallelogram has two.
  • Calling an isosceles triangle equilateral. Two equal sides make a triangle isosceles; equilateral needs all three sides equal.
  • Writing a description instead of a name. "Symmetrical" or "a triangle with two equal sides" is not a name, but isosceles is.
  • Treating a kite as if all four sides are equal. A kite has two pairs of equal sides, next to each other, so its perimeter is two lots of each length.
  • Using a fact the diagram doesn't give. Don't treat sides as equal or parallel because they look it; use only the marks and the words.

Practice questions

More on this topic: Properties of 2D Shapes worksheet with full solutions