Circle Theorems & Solids

Circle theorem questions are usually reasoning questions in disguise. You are not only calculating an angle; you are proving that a particular relationship must be true because of the way points, chords, tangents, and arcs are arranged.

Solids questions test whether you can connect a 3D object to its faces, edges, vertices, surface area, or volume. In Paper 02, diagrams may combine several ideas, so label the diagram, identify the theorem or formula, and then calculate. A clear sentence explaining the chosen rule can protect marks even if arithmetic goes wrong later.

Basic Definitions¶

Circle vocabulary tells you which theorem or formula is relevant. For example, a tangent question uses different facts from a chord or arc question.

• Arc: Part of the circumference
• Chord: Line segment joining two points on circle
• Sector: Region bounded by two radii and an arc
• Segment: Region bounded by a chord and arc
• Tangent: Line touching circle at exactly one point

Key Theorems¶

Circle theorems are relationships that are always true in a circle. In Paper 02, quote the theorem as your reason before or after calculating the angle.

Theorem 1: Angle at center is twice angle at circumference (for same arc) $\text{Angle at center} = 2 \times \text{Angle at circumference}$

Theorem 2: Angle in semicircle is 90° (Angle subtended by diameter at any point on circle is a right angle)

Theorem 3: Angles in same segment are equal (All angles subtended by same arc on same side are equal)

Theorem 4: Opposite angles in cyclic quadrilateral sum to 180° (Quadrilateral with all vertices on circle; opposite angles supplementary)

Theorem 5: Tangent perpendicular to radius (Tangent to circle ⊥ radius at point of contact)

Theorem 6: Two tangents from external point are equal (If two tangent lines from outside point touch circle, they have same length)

Theorem 7: Alternate segment theorem (Angle between tangent and chord = angle in alternate segment)

Applying Circle Theorems¶

Before calculating, identify the arc, chord, tangent, or cyclic quadrilateral involved. The diagram usually contains more information than the numbers alone.

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Part 13: Solids¶

Types of Solids¶

Solids are 3D objects, so describe them using faces, edges, vertices, surface area, and volume. A net can help you see all the faces that make up the surface.

• Prism: Two parallel identical faces (bases), with rectangular faces connecting them

  • Triangular prism, rectangular prism (cuboid), cylinder

• Pyramid: Polygonal base with triangular faces meeting at apex

  • Triangular pyramid (tetrahedron), square pyramid, etc.

• Cylinder: Two circular bases, curved side surface

• Cone: Circular base with apex connected by curved surface

• Sphere: All points equidistant from center

Euler's Formula¶

Euler's formula connects the structure of many solids. It is a quick way to check whether your count of faces, vertices, or edges is consistent.

For any polyhedron (solid with flat faces): $V - E + F = 2$

Where $V$ = vertices, $E$ = edges, $F$ = faces