Geometry

Important terms

Triangles

Quadrilaterals (four-sided shapes)

Polygons (many-sided shapes)

• Exterior angle $= 180^\circ -$ interior angle.
• Exterior angle $= 360^\circ \div$ number of sides (regular polygons only).
• Sum of interior angles $= (n - 2) \times 180^\circ$.
• Interior angle $= \dfrac{(n - 2) \times 180^\circ}{n}$ (regular polygons only).

$n = 6$, so the interior angles add up to $(6 - 2) \times 180^\circ = 720^\circ$. One angle is $720^\circ \div 6 = 120^\circ$. Check: the exterior angle is $360^\circ \div 6 = 60^\circ$, and $180^\circ - 60^\circ = 120^\circ$.

Solids (3D shapes)

Circles

Constructing triangles using a compass

If you are asked to make a triangle with sides of specific lengths, follow these steps. You need paper, a compass with a pencil and a ruler.

1. Draw a 5 cm line using a ruler.
2. Open your compass to 4 cm. Put the point on one end of the line and draw an arc above the line.
3. Open your compass to 3 cm. Put the point on the other end of the line and draw another arc that crosses the first one.
4. Draw two lines from the ends of the 5 cm line to the point where the two arcs cross. Measure the sides at the end to confirm that they are the lengths you need.

Nets

Nets are 2D shapes joined together that fold to make a 3D shape. Think of a cardboard box, such as a box that contains biscuits. Think of the net as a flat piece of cardboard before folding. After folding the net, it forms the 3D shape it is meant to.

A square with a triangle on each side folds into a pyramid (a square base and sides of triangles).

Two triangles joined by three rectangles fold into a triangular prism.

The prism has a right-angled triangle at each end, with sides 6, 8 and 10, and it is 12 long.

Scale is used to adjust drawings. Here the drawing would be too big, so we make it smaller by halving all the lengths. The net has five faces:

Draw the three rectangles side by side, sharing their 6 cm sides, and attach one triangle to each end of the $4 \times 6$ rectangle (along its 4 cm sides).

Similarity

When two shapes are similar, they have the same shape but different sizes. Their angles are equal, and their sides are in the same ratio.

The scale factor here is 2, because all the sides doubled in length.

When two shapes (like cubes or spheres) are similar and the lengths increase by scale factor $k$:

• Lengths increase by $k$.
• Areas increase by $k^2$.
• Volumes increase by $k^3$.

The 2 cm base edge matches the 6 cm base edge, and the 4 cm edge matches the missing length $x$:

$\frac{6}{2} = \frac{x}{4} \quad\Rightarrow\quad x = 12\text{ cm}$

The length scale factor is $\frac{10}{5} = 2$, so the volume scale factor is $2^3 = 8$. The larger volume is $40 \times 8 = 320$ cm³.

Symmetry

Line symmetry (also called mirror symmetry): a shape has line symmetry if you can fold it and both halves match perfectly.

Rotational symmetry: a shape has rotational symmetry if it looks the same when you turn it (by less than a full circle). The number of times it looks the same in one full turn is the order of rotational symmetry. A shape that only looks the same after a full turn has order 1 (no rotational symmetry).

• The equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3.
• The kite has 1 line of symmetry and rotational symmetry of order 1.

Here is a $4 \times 4$ grid. A square is shaded (■) where it is grey in the original exercise: row 1, columns 3 and 4; row 3, column 3; and row 4, column 1.

Rotational symmetry of order 2 means the shape looks the same after a half turn ($180^\circ$) about the centre of the grid. A half turn sends the square in row $r$, column $c$ to row $5 - r$, column $5 - c$. So:

• row 1, column 3 goes to row 4, column 2, which is not shaded yet;
• row 1, column 4 goes to row 4, column 1, which is shaded;
• row 3, column 3 goes to row 2, column 2, which is not shaded yet;
• row 4, column 1 goes to row 1, column 4, which is shaded.

So shade row 2, column 2 and row 4, column 2.

Angles

Angle rules at a point, on a straight line and between lines

$w + x + y + z = 360^\circ$

$x + y + z = 180^\circ$

Angles in parallel lines

Corresponding angles (F-shaped) are equal. Arrows on lines mean that the lines are parallel.

Alternate angles (Z-shaped) are equal. Also, the angles between the parallel lines on the same side (co-interior angles) add up to $180^\circ$.

$a = b,\quad x = y,\quad a + x = 180^\circ,\quad y + b = 180^\circ$

Circle properties

Angle in a semicircle $= 90^\circ$

When you make a triangle by connecting the ends of a diameter to a point on the circle, it will always be a right-angled triangle.

Angle between a tangent and a radius $= 90^\circ$

The angle that a tangent makes with the radius at the point of contact will always be $90^\circ$.

Angle at the centre $= 2 \times$ angle at the circumference

The angle from a chord at the centre is twice that at the circumference of the circle.

Angles in the same segment are equal

Similar to a semicircle, all the angles drawn from the same chord on the same side of it will be equal.

Opposite angles in a cyclic quadrilateral add to $180^\circ$

If all the corners of a quadrilateral touch the circumference of a circle, the angles on opposite corners add up to $180^\circ$.

$a + b = 180^\circ \qquad x + y = 180^\circ$

Alternate segment theorem

When the corner of a triangle touches the tangent of a circle, the angle between the tangent and a chord equals the angle in the alternate (opposite) segment.

$x = y \qquad a = b$

Equal chords are equidistant from the centre

If two chords in a circle are the same length, they are the same distance from the centre.

The perpendicular bisector of a chord passes through the centre

If you draw a perpendicular line that cuts a chord exactly in half, it will go through the centre of the circle.

Tangents from an external point are equal in length

If you draw two tangents from the same point outside a circle, they will both be the same length.