Jupiter2U School
Chapter 11

11. Geometry

High School

Geometry reasons about shape, size and position. From a handful of axioms flow the theorems that measure the world: angles, triangles, the Pythagorean theorem, area and volume.

At a glance
Core ideaFrom a few axioms, deductive proof derives truths that hold for every shape at once.
Key termPythagorean theorem: a² + b² = c² in any right triangle.
You can…Find lengths, prove congruence and similarity, and compute area and volume.
Watch outArea scales with the square of length — double the radius, quadruple the area.
01 · Theory

Angles, triangles and Pythagoras

Geometry starts from undefined primitives (point, line, plane) and axioms. Core angle facts: angles on a straight line sum to 180°, angles around a point to 360°, and vertically opposite angles are equal. When a line crosses two parallels, corresponding and alternate angles are equal.

For any triangle the interior angles sum to 180°. In a right triangle, the Pythagorean theorem relates the sides:

a² + b² = c²   (c is the hypotenuse)

Triangles are congruent (identical) under conditions SSS, SAS, ASA or RHS, and similar (same shape, scaled) when their angles match — then corresponding sides are in constant ratio. Key measures:

Circle:   C = 2πr    A = πr²
Triangle area:   A = ½ · base · height
Cylinder volume:   V = πr²h
02 · Explanation

Why proof, not just measurement

You could measure a thousand triangles and find their angles near 180°, but measurement has error and never covers every triangle. Geometry's power is deductive proof: from agreed axioms you derive a truth that holds for all cases at once, exactly. That is why the angle-sum result is certain, not merely likely.

The Pythagorean theorem is the hinge of measurement — it converts the abstract idea of "distance" into arithmetic on coordinates, and later becomes the foundation of trigonometry and the distance formula. Similarity is the everyday superpower: it is how a small shadow lets you compute a tall building's height, because matching angles force matching side-ratios regardless of scale.

Perimeter

  • The distance around a shape's edge.
  • Measured in length units (cm, m).
  • Add every side — a rectangle is 2(l + w).
  • Doubling the size doubles the perimeter.

Area

  • The space inside a shape.
  • Measured in square units (cm², m²).
  • Multiply dimensions — a rectangle is l × w.
  • Doubling the size quadruples the area.
03 · Practical

Worked example: ladder and shadow

Problem A — Pythagoras

A ladder 5 m long leans against a wall with its foot 3 m out. How high up the wall does it reach?

  1. The ladder is the hypotenuse: c = 5, base b = 3, height a = ?.
  2. Rearrange: a² = c² − b² = 25 − 9 = 16.
  3. a = √16 = 4 m. (A classic 3-4-5 triangle.)
3² = 9 4² = 16 3 m 4 m 5² = 25 = 9 + 16 ladder = 5 m (hypotenuse)
The ladder, wall and ground form a 3-4-5 right triangle: 3² + 4² = 5², so the ladder reaches 4 m up the wall.
Problem B — similar triangles

A 2 m person casts a 3 m shadow; a nearby tree casts a 24 m shadow at the same time. How tall is the tree?

  1. Sun angle is shared, so the triangles are similar: height / shadow is constant.
  2. 2/3 = H/24, so H = 24 × 2/3 = 16 m.
04 · Q&A

Test your understanding

Does a 6, 8, 11 triangle contain a right angle?

Test the converse of Pythagoras on the longest side: 6² + 8² = 36 + 64 = 100, but 11² = 121. Since 100 ≠ 121, it is not right-angled. Because 121 > 100, the angle opposite the 11-side is actually obtuse.

Why do all triangles have angles summing to exactly 180°?

Draw a line through one vertex parallel to the opposite side. The two base angles reappear at that vertex as alternate angles, and together with the top angle they lie on a straight line — which is 180°. So the three interior angles must total 180°. (This relies on the parallel postulate; on a curved surface the sum differs.)

Two circles have radii 3 and 6. How do their areas compare?

Area scales with the square of the radius: A = πr². Doubling the radius multiplies area by 2² = 4. So the larger circle has four times the area (π·36 vs π·9), not twice. This square-law is why length, area and volume scale differently.

What is the length of the diagonal of a rectangle 8 by 15?

The diagonal is the hypotenuse of a right triangle with legs 8 and 15: d = √(8² + 15²) = √(64 + 225) = √289 = 17. This is the 8-15-17 Pythagorean triple, and it is exactly the 2-D distance formula in action.

Concept mind map

How the ideas connect

Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.

AnglesTriangle typesAngle sum 180Pythagoras theoremCongruenceProof not measureGeometry and Pythagoras
Infographic

The key facts, visualised

180 deg
Angles in a triangle sum to this
a^2+b^2=c^2
Pythagoras for right triangles
360 deg
Angles around a point
Hypotenuse
Longest side, opposite right angle
Solved examples

Worked problems, step by step

Follow each solution line by line, then try to reproduce it on paper before moving on.

Example 1A right triangle has legs 3 and 4. Find the hypotenuse.

  1. Use a^2 + b^2 = c^2
  2. 3^2 + 4^2 = 9 + 16 = 25
  3. c = sqrt(25) = 5

Example 2Two angles of a triangle are 50 and 60 degrees. Find the third.

  1. Angles sum to 180
  2. 180 - 50 - 60 = 70
  3. Third angle is 70 degrees
Practice problem set

Now you try

Work each one out first, then tap to reveal the worked answer.

1A ladder reaches 12 m up a wall with base 5 m out. Find its length.
sqrt(12^2 + 5^2) = sqrt(169) = 13 m.
2Find the missing leg if hypotenuse 10 and one leg 6.
sqrt(100 - 36) = sqrt(64) = 8.
3What are the base angles of an isosceles triangle with apex 40 deg?
The remaining 140 splits equally, so each is 70 deg.
4Is a triangle with sides 5, 12, 13 right-angled?
Yes, since 25 + 144 = 169 = 13^2.
5Two angles on a straight line are 110 and x. Find x.
They sum to 180, so x = 70 deg.
6A square has side 2. Find its diagonal.
Diagonal = sqrt(2^2 + 2^2) = 2 sqrt(2).