2.8 Angles, Coordinates, 3D Shape & Volume

Angles, Coordinates, 3D Shape & Volume

Geometry in the Kent Test maths section is diagram-led. The question rarely says "use the angle sum of a triangle"; it shows a shape with two angles labelled and a third marked with a letter. Success comes from recognising which of a small set of rules applies, then applying it once or twice.

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The Three Angle Rules That Cover Most Questions

Add the vocabulary, because questions ask for it by name: acute is under 90°, right is exactly 90°, obtuse is between 90° and 180°, reflex is over 180°, and a full turn is 360°.

Parallel lines

When a straight line crosses two parallel lines:

• Corresponding angles (F-shape) are equal.
• Alternate angles (Z-shape) are equal.
• Co-interior angles (C-shape) add to 180°.

Two parallel lines are crossed by a transversal. One angle measures 68°. What is the co-interior angle on the same side? 180 − 68 = 112°.

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Angles Inside Shapes

• Triangle: angles total 180°.
• Quadrilateral: angles total 360°.
• Any polygon with n sides: interior angles total (n − 2) × 180°. A pentagon gives 540°, a hexagon 720°.
• Regular polygon: every interior angle is the total divided by n. A regular hexagon has 720 ÷ 6 = 120° at each vertex.
• Exterior angles of any polygon always total 360°, so a regular polygon's exterior angle is 360 ÷ n.

Triangle types are examinable by their angle and side properties:

An isosceles triangle has an apex angle of 40°. Find each base angle. 180 − 40 = 140, shared equally: 70° each.

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Coordinates in Four Quadrants

Coordinates are written (x, y) and read across first, then up. The phrase "along the corridor, then up the stairs" fixes the order permanently.

The Kent Test uses all four quadrants, so both values can be negative:

• Top right: (+, +) | Top left: (−, +) | Bottom left: (−, −) | Bottom right: (+, −)

Translation is described in steps: "translate 3 right and 2 down" changes (−1, 4) to (2, 2). Adding to x moves right, subtracting moves left; adding to y moves up, subtracting moves down.

Finding a missing vertex is a standard item: three corners of a rectangle are given as (1, 2), (5, 2) and (1, 6). The fourth must share an x-value with (5, 2) and a y-value with (1, 6), so it is (5, 6).

Midpoints are the average of each coordinate: the midpoint of (2, 3) and (8, 9) is ((2+8)/2, (3+9)/2) = (5, 6).

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3D Solids: Faces, Edges, Vertices

A prism has the same cross-section all the way along; a pyramid tapers to a single point. Confusing the two is a common source of wrong face counts.

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Volume and Surface Area of a Cuboid

• Volume = length × width × height, measured in cubic units (cm³, m³).
• Surface area = the total of six faces, which come in three matching pairs: 2(lw) + 2(lh) + 2(wh), measured in square units.

A box is 8 cm by 5 cm by 3 cm. Volume = 8 × 5 × 3 = 120 cm³. Surface area = 2(40) + 2(24) + 2(15) = 80 + 48 + 30 = 158 cm².

Capacity link: 1 cm³ holds exactly 1 ml, and 1,000 cm³ holds 1 litre. So the box above holds 120 ml. That conversion turns volume questions into capacity questions and back again, and it appears regularly.

Counting cubes in a stack

Where a maths question shows cubes stacked into a complete rectangular block, do not count the visible faces: multiply the layers. A 4 × 3 base with 2 layers is 4 × 3 × 2 = 24 cubes, even though far fewer are drawn. Incomplete stacks with gaps behind the visible cubes are a spatial reasoning task rather than an arithmetic one, and are covered in section 4.4.

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Working Under the Clock

Geometry items reward a fixed opening move. Mark every known angle on the diagram in the booklet before doing any arithmetic — rough work in the booklet is allowed and is not marked, so there is no reason to hold the numbers in your head. Once the diagram is annotated, the rule to use is usually obvious, and the calculation is a single subtraction.