Geometry

Angles, triangles, polygons, circles, solids and coordinates. Figures may not be drawn to scale; trust the numbers, not the picture.

Angles

Angles on a straight line

Adjacent angles on a line add up to 180° (supplementary).

Vertical angles

Where two lines cross, opposite angles are equal and neighbours add to 180°.

Parallel lines

A line crossing two parallels makes only two kinds of angles: all acute ones are equal, all obtuse ones are equal, and acute + obtuse = 180°.

Angles of a triangle

The angles of any triangle add up to 180°.

The exterior angle of a triangle

Exterior angle = the sum of the two far (remote) interior angles.

Isosceles triangle

Equal sides face equal angles.

Interior angles of a polygon

Sum = (n − 2) × 180°; each angle of a regular one = sum ÷ n.

Exterior angles of a polygon

Exterior angles of any polygon always total 360°, so each one of a regular polygon = 360 ÷ n.

Diagonals of a polygon

diagonals = n(n − 3) 2

Triangles

Pythagoras

In a right triangle: hypotenuse² = the sum of the other two squared; the hypotenuse faces the right angle.

Diagonal of a square or rectangle

The diagonal is the hypotenuse of a right triangle made by the length and width. In a square: diagonal = side × √2.

Special right triangles

30-60-90: sides 1: √3: 2; 45-45-90: sides 1: 1: √2.

Triangle inequality

The third side is more than the difference and less than the sum.

Similar triangles

Find the scale factor from two matching sides, then multiply.

Area of a triangle

½ × base × height, the height is perpendicular to the base.

Quadrilaterals and circles

Rectangle and square

Rectangle: area = l × w, perimeter = 2(l + w). Square: area = s², diagonal = s√2.

Rhombus, trapezoid, parallelogram

Rhombus: ½ × d1 × d2. Trapezoid: ½ × (sum of parallel sides) × height. Parallelogram: base × height.

The circle

Area = πr², circumference = 2πr, diameter = 2r.

Sector and arc

Take a fraction of the circle = angle ÷ 360, then multiply by the area or circumference.

Shaded fractions of a circle

Count the equal parts: fraction = shaded ÷ total. One part’s angle = 360 ÷ the number of parts, and its area = the area ÷ the number of parts.

Sector angle and its fraction

Fraction = angle ÷ 360, and angle = fraction × 360.

Circumference, area and wheels

From any given, find the radius first. A wheel: distance = turns × circumference.

Shaded fraction of a grid

Number of squares = rows × columns; fraction = shaded ÷ total, then simplify.

Inscribed and central angles

Inscribed angle = half the central angle on the same arc.

Shaded areas

Shaded = big shape − small shape.

Solids and coordinates

Cube and box

Volume = l × w × h. Cube: volume s³, surface area 6s².

Cylinder

Volume = base area × height = πr²h.

Scaling a figure

If lengths are multiplied by k: perimeter × k, area × k², volume × k³. A 10% longer side gives 21% more area, not 20%.

Distance between points

Draw a right triangle: the legs are the x-difference and y-difference, then Pythagoras.

Midpoint and slope

Midpoint = average of the coordinates. Slope = rise ÷ run.

Diameter = 2 × radius; never plug the diameter into the area formula.
In a parallelogram use the height, not the slanted side.

Hard triangles and angles

Pythagorean families

Learn (3, 4, 5) and its multiples (6, 8, 10 … 30, 40, 50), plus (5, 12, 13), (8, 15, 17) and (7, 24, 25). Spot two sides of a family and the third is known without working.

Triangle type from its sides

Compare the longest side squared with the other two squared and added: equal → right, smaller → acute, bigger → obtuse.

Angles written with x

Vertical angles → equal. On a line → 180. Around a point → 360. Complementary → 90. In a triangle → 180. Write the equation, then solve.

The bent angle between parallels

The bend angle between two parallels = the sum of the two angles at the parallels. Cyclic quadrilateral: opposite angles add to 180. An angle standing on a diameter = 90.

Special and equilateral triangles

30-60-90: x, x√3, 2x. 45-45-90: x, x, x√2. Equilateral: height = (√3/2) × side, area = (√3/4) × side².

Midpoints, parallels, angle bisectors and scale

The midpoint segment is half the third side. A line parallel to the base cuts both sides in the same ratio. An angle bisector splits the opposite side in the ratio of the two neighbouring sides. A 1: k scale multiplies areas by k².

Counting rectangles, squares, tiles; the staircase

Rectangles = (1 + … + columns) × (1 + … + rows). Squares in an n × n grid = 1² + 2² + … + n². Staircase perimeter = 2 × (base + height).

Cone, sphere, cube, ring and laps

Cone = ⅓ πr²h. Sphere = ⁴⁄₃ πr³. Cube: total 6a², lateral 4a². Ring = π(R² − r²). Laps = number of laps × 2πr.

Matchsticks, cutting a circle, shapes inside shapes, and the leaf

Sticks: n squares in a row = 3n + 1, n triangles = 2n + 1. Circle parts: diameters through the centre = 2n, chords from one point = n + 1, most parts = n² + n + 2 2. Inside shapes: square in circle: its diagonal is the diameter; circle in square: diameter = side; hexagon side = radius; equilateral triangle side = r 3.

The regular hexagon, a square from its diagonal, a rectangle in a circle, circles on a diameter, shaded areas

Hexagon: six equilateral triangles; long diagonal = 2 × side, area = 3 2 √3 × side². Square: area = ½ diagonal². Rectangle with corners on a circle: its diagonal is the diameter. n circles along a diameter: circumference ratio 1 n, area ratio 1 n². A triangle on a rectangle's side with its tip on the opposite side is half the rectangle; the midpoint square is half the square.

Triangles between parallels, the angle bisector, a tank in litres, a cylinder's surface

Between parallel lines the height is shared, so areas are in the ratio of the bases. The angle bisector splits the opposite side in the ratio of the sides: AB AC = BD DC. 1000 cm³ = 1 litre, 1 m³ = 1000 litres. Cylinder: curved area = 2πrh; total = curved + 2πr².