Geometry and Measures

The One Thing

Geometry and measures problems are solved by identifying the relevant geometric property, theorem, construction or formula, then substituting accurately with consistent units. Careful diagrams, exact values and appropriate checking are essential when determining lengths, angles, areas, volumes, positions and movements.

Definitions and Results

• Angle properties: Angles on a straight line add to $180^{\circ }$; angles around a point add to $360^{\circ }$; vertically opposite angles are equal. Angles in a triangle add to $180^{\circ }$, and angles in a quadrilateral add to $360^{\circ }$.
• Parallel lines: When lines are parallel, corresponding angles and alternate angles are equal, while co-interior angles add to $180^{\circ }$.
• Polygon: A polygon is a closed two-dimensional shape with straight sides. The interior angle sum of an $n$-sided polygon is $(n-2)\times 180^{\circ }$.
• Regular polygon: The exterior angle is $360^{\circ }÷n$, and each interior angle is $180^{\circ }-360^{\circ }÷n$.
• Circle theorems: The angle in a semicircle is $90^{\circ }$; angles in the same segment are equal; and opposite angles in a cyclic quadrilateral add to $180^{\circ }$.
• Tangent to a circle: A tangent is perpendicular to the radius at the point of contact. The angle between a tangent and a chord equals the angle in the alternate segment.
• Similarity: Similar shapes have equal corresponding angles and proportional corresponding lengths. For scale factor $k$, lengths multiply by $k$, areas by $k^2$, and volumes by $k^3$. A scale factor greater than $1$ enlarges a shape, while a scale factor between $0$ and $1$ reduces it.
• Congruence: Congruent shapes have exactly the same size and shape, even if one has been rotated, reflected or translated.
• Geometric construction: Accurate diagrams made using a ruler and compass, including perpendicular bisectors, angle bisectors, perpendiculars and loci.
• Locus: The set of all points satisfying a given condition, such as being a fixed distance from a point or equidistant from two points.
• Perimeter and area: Perimeter measures the boundary of a two-dimensional shape; area measures the space inside it.
• Circle measures: Circumference is $2\pi r$ or $\pi d$, and area is $\pi r^2$. Arc length and sector area are fractions of the circumference and circle area.
• Surface area and volume: Surface area measures the outside of a three-dimensional object; volume measures the space it occupies.
• Trigonometry: In a right-angled triangle,

$$\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}}.$$

• Pythagoras’ theorem: For a right-angled triangle,

$$a^2+b^2=c^2,$$ where $c$ is the hypotenuse.

• Exact trigonometric values:

$$\sin 30^{\circ }=\frac{1}{2},\cos 60^{\circ }=\frac{1}{2},\tan 45^{\circ }=1,\sin 45^{\circ }=\frac{\sqrt{2}}{2},\cos 45^{\circ }=\frac{\sqrt{2}}{2}.$$

• Vector: A quantity with both size and direction, commonly written as a column vector or directed line segment.
• Vector addition and subtraction: Vectors are combined component by component. Reversing a vector changes its direction and gives its negative.
• Column vector: A vector written vertically, such as $\left[\begin{array}{c}a\\ b\end{array}\right]$, representing a horizontal movement of $a$ and a vertical movement of $b$.
• Bearings: Bearings are measured clockwise from north and written as three-figure angles, such as $065^{\circ }$.
• Compound measures: Important examples include

$$\text{speed}=\frac{\text{distance}}{\text{time}},\text{density}=\frac{\text{mass}}{\text{volume}},\text{pressure}=\frac{\text{force}}{\text{area}}.$$

Worked Methods

Applying angle properties

• Identify the relevant line, point, triangle, quadrilateral or pair of parallel lines.
• Use the appropriate result:

- straight line: $180^{\circ }$; - point: $360^{\circ }$; - triangle: $180^{\circ }$; - quadrilateral: $360^{\circ }$; - vertically opposite angles: equal; - corresponding or alternate angles in parallel lines: equal; - co-interior angles in parallel lines: $180^{\circ }$.

• Form and solve the resulting equation.

For a regular $n$-sided polygon, calculate the exterior angle using $360^{\circ }÷n$, then calculate each interior angle using $180^{\circ }-360^{\circ }÷n$.

Using circle theorems

• Identify whether the diagram contains a semicircle, a cyclic quadrilateral, equal angles in the same segment, a tangent, or a chord.
• Apply the relevant theorem:

- angle in a semicircle $=90^{\circ }$; - angles in the same segment are equal; - opposite angles in a cyclic quadrilateral add to $180^{\circ }$; - a tangent is perpendicular to the radius at the point of contact; - the angle between a tangent and a chord equals the angle in the alternate segment.

• Combine the theorem with angle properties if further angles are required.

Using similarity and scale factors

• Match corresponding angles and corresponding sides.
• Determine the length scale factor $k$.
• Multiply lengths by $k$, areas by $k^2$, and volumes by $k^3$.
• State whether the transformation enlarges the shape if $k>1$, or reduces it if $0<k<1$.

Performing geometric constructions and loci

• Use a ruler and compass accurately.
• For a perpendicular bisector, draw equal-radius arcs from each endpoint and join the arc intersections.
• For an angle bisector, draw an arc from the angle vertex, then draw equal-radius arcs from the two points where it meets the arms and join the resulting intersection to the vertex.
• For a perpendicular, construct the required equal-radius arcs and join their intersections.
• For a locus, identify the condition:

- a fixed distance from a point produces a circle; - equidistance from two points produces a perpendicular bisector.

• Show all relevant arcs and lines clearly.

Calculating perimeter and area

• Label the relevant lengths and identify the shape.
• Select the appropriate formula:

$$\text{area of triangle}=\frac{1}{2}\times \text{base}\times \text{perpendicular height},$$ $$\text{area of parallelogram}=\text{base}\times \text{perpendicular height},$$ $$\text{area of trapezium}=\frac{1}{2}\times (\text{sum of parallel sides})\times \text{perpendicular distance}.$$

• Substitute values using consistent units.
• Give the area in squared units and the perimeter in linear units.

Calculating circle measures

• Identify the radius $r$ or diameter $d$.
• Use

$$\text{circumference}=2\pi r=\pi d$$ or $$\text{area}=\pi r^2.$$

• For an arc length or sector area, calculate the relevant fraction of the full circumference or circle area.
• Retain $\pi$ exactly unless a decimal approximation is requested.

Calculating surface area and volume

• Identify the three-dimensional object and label its dimensions.
• Use the appropriate formula:

$$\text{prism volume}=\text{cross-sectional area}\times \text{length},$$ $$\text{cuboid volume}=\text{length}\times \text{width}\times \text{height},$$ $$\text{cylinder volume}=\pi r^{2}h,$$ $$\text{cylinder curved surface area}=2\pi rh,$$ $$\text{cone volume}=\frac{1}{3}\pi r^{2}h,$$ $$\text{sphere volume}=\frac{4}{3}\pi r^3,$$ $$\text{sphere surface area}=4\pi r^2.$$

• Include all required faces or curved surfaces when finding total surface area.
• Give volume in cubed units and surface area in squared units.

Solving right-angled triangles

• Identify the chosen angle.
• Label the sides relative to that angle as opposite, adjacent and hypotenuse.
• Use the relevant trigonometric ratio:

$$\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}}.$$

• Rearrange or use inverse sine, cosine or tangent to find an unknown angle.
• Check that the calculator is in degree mode when working with degrees.
• Alternatively, use Pythagoras’ theorem,

$$a^2+b^2=c^2,$$ when the triangle is right-angled and the relevant side lengths are known.

Using the sine rule, cosine rule and triangle-area formula

• Identify the known sides and angles.
• Use the sine rule when a side-angle opposite pair is available:

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$$

• Use the cosine rule when two sides and the included angle, or three sides, are given:

$$a^2=b^2+c^2-2bc\cos A.$$

• Use the area formula when two sides and their included angle are known:

$$\text{area}=\frac{1}{2}ab\sin C.$$

• Retain exact forms where appropriate and round only at the end.

Working with vectors

• Represent each movement as a column vector, such as

$$\left[\begin{array}{c}a\\ b\end{array}\right],$$ where $a$ is horizontal movement and $b$ is vertical movement.

• Add or subtract corresponding components.
• Reverse a vector by changing both component signs.
• Use vectors to describe translations and to prove relationships such as parallelism, equal lengths and points dividing line segments.

Working with bearings and measures

• Measure the bearing clockwise from north.
• Write the result as a three-figure angle, such as $065^{\circ }$.
• For compound measures, select and apply the appropriate ratio:

$$\text{speed}=\frac{\text{distance}}{\text{time}},\text{density}=\frac{\text{mass}}{\text{volume}},\text{pressure}=\frac{\text{force}}{\text{area}}.$$

• Check that the units are compatible before calculating.

Where It Goes Wrong

• Selecting a theorem or formula without first identifying the relevant information, such as corresponding sides, parallel lines, right angles or the included angle.
• Confusing the opposite, adjacent and hypotenuse sides in trigonometry; these must be identified relative to the chosen angle.
• Applying Pythagoras’ theorem or right-angled trigonometric ratios to a triangle that is not right-angled.
• Forgetting the conditions of circle theorems, parallel-line angle rules, similarity scale factors or tangent-radius perpendicularity.
• Omitting perpendicular height in the area of a triangle, parallelogram or trapezium, or omitting curved surfaces when calculating surface area.
• Using incorrect units, rounding intermediate values too early, failing to retain exact forms such as $\pi$ and surds, or leaving a calculator in the wrong mode instead of degree mode.

What Gets Asked

• Calculate unknown angles using straight-line, point, triangle, quadrilateral, polygon, parallel-line and circle-theorem properties.
• Find exterior and interior angles of regular polygons.
• Use tangents, chords, cyclic quadrilaterals, semicircles and angles in the same segment.
• Determine missing lengths, areas, surface areas and volumes using geometric formulas.
• Calculate circumference, area, arc length and sector area.
• Use similarity, congruence and scale factors to compare lengths, areas and volumes.
• Construct perpendicular bisectors, angle bisectors, perpendiculars and loci using a ruler and compass.
• Solve right-angled triangle problems using Pythagoras’ theorem, sine, cosine and tangent.
• Use exact trigonometric values, inverse trigonometric functions, the sine rule, the cosine rule and $\frac{1}{2}ab\sin C$.
• Interpret and combine column vectors, including vector addition, subtraction and reversal.
• Solve bearing problems and problems involving speed, density and pressure.
• Select appropriate formulas, maintain consistent units, retain exact values and round final answers to the requested accuracy.