Geometry begins with the building blocks: points (exact locations in space), line segments (two endpoints connected by a straight path), rays (a starting point going infinitely in one direction), and lines (extending infinitely in both directions).
When two rays share an endpoint — called the vertex — they form an angle. Angles are measured in degrees (°). A full rotation is 360°.
The Four Basic Angle Types
0°–89°
Acute
Less than 90° — "a cute little angle"
90°
Right
Exactly 90° — marked with a small square
91°–179°
Obtuse
Greater than 90° but less than 180°
180°
Straight
A straight line — both rays point opposite ways
Complementary vs Supplementary: Complementary angles add up to 90°. Supplementary angles add up to 180°. Memory trick: Complementary = corner (right angle), Supplementary = straight line.
Problem: Two angles are supplementary. One angle is 65°. Find the other.
Supplementary angles sum to 180°.
x + 65° = 180°
x = 180° - 65°
x = 115°
The other angle is 115° (obtuse).
Part 2
Triangles: Classification and Properties
A triangle is a polygon with three sides and three angles. One crucial rule: the three angles of any triangle always add up to 180°. This rule alone lets you find a missing angle in any triangle.
Classifying by Angles
Acute triangle — all three angles are less than 90°
Right triangle — one angle is exactly 90°
Obtuse triangle — one angle is greater than 90°
Classifying by Sides
Equilateral — all three sides equal, all angles equal 60°
Isosceles — two sides equal, two base angles equal
Scalene — all three sides different lengths, all angles different
Problem: A triangle has angles of 50° and 70°. What is the third angle?
Rule: All angles sum to 180°.
50° + 70° + x = 180°
120° + x = 180°
x = 60°
The third angle is 60°.
Since all angles are less than 90°, this is an ACUTE triangle.
Part 3
Area and Perimeter Formulas
Perimeter = the total distance around the outside of a shape. Think of fencing a garden — you need to know the total length of fence. Units are linear (cm, m, ft).
Area = the amount of flat space inside a shape. Think of painting a wall — you need to know how much surface to cover. Units are squared (cm², m², ft²).
Shape
Area Formula
Perimeter Formula
Rectangle
A = l × w
P = 2(l + w)
Square
A = s²
P = 4s
Triangle
A = ½ × b × h
P = a + b + c
Circle
A = π × r²
C = 2πr (Circumference)
Parallelogram
A = b × h
P = 2(a + b)
Trapezoid
A = ½(b₁ + b₂) × h
P = sum of all sides
Pi (π) ≈ 3.14159 — Pi is the ratio of a circle's circumference to its diameter. It's the same for every circle, no matter the size. In exams, use π ≈ 3.14 unless told otherwise.
Worked Example: Rectangle
A bedroom measures 5 m long and 4 m wide.
Find: (a) area and (b) perimeter.
(a) Area = l × w = 5 × 4 = 20 m²
(b) Perimeter = 2(l + w) = 2(5 + 4) = 2 × 9 = 18 m
Note the units: area is m² (square metres), perimeter is m (metres).
Worked Example: Triangle
A triangle has base = 8 cm and perpendicular height = 5 cm.
Find its area.
Area = ½ × b × h
= ½ × 8 × 5
= ½ × 40
= 20 cm²
IMPORTANT: The height must be perpendicular (at 90°) to the base.
Worked Example: Circle
A circular pizza has a radius of 14 cm.
Find: (a) area and (b) circumference. Use π ≈ 3.14.
(a) Area = π × r² = 3.14 × 14² = 3.14 × 196 = 615.44 cm²
(b) Circumference = 2πr = 2 × 3.14 × 14 = 87.92 cm
Radius = half the diameter. If given diameter, divide by 2 first!
Part 4
The Pythagorean Theorem
The Pythagorean theorem is one of the most famous equations in all of mathematics. It applies specifically to right triangles (triangles with a 90° angle).
a² + b² = c²
Where:
a and b are the two legs (the sides that form the right angle)
c is the hypotenuse — the longest side, always opposite the right angle
How to identify the hypotenuse: Look for the right angle marker (a small square) in the triangle. The side directly across from that right-angle corner — not touching it — is the hypotenuse.
Label the sides
Identify which side is the hypotenuse (c) — it's opposite the 90° angle and always the longest side. Label the other two sides a and b.
Substitute into the formula
Replace a, b, and c with the known values. If finding c, substitute a and b. If finding a leg, substitute the leg you know and c.
Square the known values
Calculate a² and b² (or whichever values you have). Remember: squaring means multiplying a number by itself. 6² = 36.
Solve and take the square root
Add or subtract to find the missing squared value, then take the square root (√) to find the actual side length.
Example 1 — Find the hypotenuse:
A right triangle has legs a = 3 cm and b = 4 cm.
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²
c = √25 = 5 cm
The hypotenuse is 5 cm. (3-4-5 is a classic "Pythagorean triple"!)
Example 2 — Find a missing leg:
A ladder 10 m long leans against a wall.
Its base is 6 m from the wall. How high up the wall does it reach?
a² + b² = c² (c = ladder = 10, b = base = 6)
a² + 6² = 10²
a² + 36 = 100
a² = 100 - 36
a² = 64
a = √64 = 8 m
The ladder reaches 8 m up the wall.
Part 5
Volume of Basic 3D Solids
Volume measures how much 3D space is inside an object. Units are cubed (cm³, m³, ft³). Think of filling a fish tank — volume tells you how much water it holds.
Solid
Volume Formula
Key Measurements
Cube
V = s³
s = side length
Rectangular prism (cuboid)
V = l × w × h
length, width, height
Cylinder
V = π × r² × h
r = radius of base, h = height
Cone
V = ⅓ × π × r² × h
r = radius of base, h = height
Sphere
V = (4/3) × π × r³
r = radius
A fish tank is a rectangular prism.
Dimensions: 60 cm long, 30 cm wide, 40 cm tall.
Find its volume.
V = l × w × h
V = 60 × 30 × 40
V = 72,000 cm³
To convert cm³ to litres: divide by 1,000.
72,000 ÷ 1,000 = 72 litres.
The fish tank holds 72 litres of water.
Test Yourself
Practice Problems
Problem 1: A triangle has two angles of 45° and 90°. What is the third angle? What type of triangle is it?
Third angle = 180° - 45° - 90° = 45°
The triangle has a 90° angle → it's a RIGHT triangle.
The two base angles are both 45° → it's also ISOSCELES.
This is a right isosceles triangle.
Problem 2: A rectangle has a length of 12 cm and a width of 7 cm. Find its area and perimeter.
Area = l × w = 12 × 7 = 84 cm²
Perimeter = 2(l + w) = 2(12 + 7) = 2 × 19 = 38 cm
Problem 3: A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse.
What is the Pythagorean theorem and when do I use it?
The Pythagorean theorem (a² + b² = c²) applies to right triangles. Use it whenever you know two sides of a right triangle and need to find the third. It's essential for finding diagonal distances, roof heights, ramp lengths, and much more in real life.
What is the difference between area and perimeter?
Perimeter is the total distance around the outside edge of a shape — like measuring how much fencing you need. Area measures the flat surface inside a shape — like how much carpet you need for a room. Perimeter uses linear units (m, cm); area uses squared units (m², cm²).
Why do the angles of a triangle always add up to 180°?
This is a fundamental property of Euclidean (flat) geometry. You can prove it visually: tear off the three corners of any triangle and line them up — they will always form a straight angle of exactly 180°. This rule lets us find missing angles in any triangle.
What is Pi and why does it appear in circle formulas?
Pi (π ≈ 3.14159) is the ratio of any circle's circumference to its diameter. This ratio is the same for every circle ever drawn — a mathematical constant of the universe. Because every circle calculation involves this relationship between its diameter and boundary, π appears in both the area formula (πr²) and the circumference formula (2πr).
What are complementary and supplementary angles?
Complementary angles add up to 90° (a right angle). Supplementary angles add up to 180° (a straight line). Memory trick: C comes before S in the alphabet, just like 90 comes before 180. So Complementary = 90°, Supplementary = 180°.
Math Fundamentals Complete! 🎉
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