Mathematics · Foundational · 30 min · Chapter 3 of 28

Geometry

Shape, measurement, and the first real proofs: congruence, similarity, and the Pythagorean theorem.

Read first: Algebra

The idea

Euclid’s geometry is a theory of space built from a few postulates. You do not need the whole axiomatic machine to use it, but you should feel the difference between a measurement and a reason. Two triangles are congruent when a correspondence of vertices makes corresponding sides and angles equal — SAS, ASA, SSS, and the right-triangle case HL are the standard warrants. Similarity is the same idea after a scale factor: corresponding angles equal, and corresponding sides proportional.

The Pythagorean theorem is the workhorse. If a right angle sits between legs a and b, the hypotenuse c satisfies a² + b² = c². The converse is also true and is how you test a triangle for a right angle. Scaling matters more than people expect: if every length in a figure is multiplied by k, areas multiply by k² and volumes by k³. Doubling a recipe of linear size multiplies the paint by four and the clay by eight.

Circles introduce π, the ratio of circumference to diameter, the same number for every circle. Area is πr², not πd, and not 2πr. A sphere’s surface is 4πr² and its volume is (4/3)πr³. Those two formulas are consistent with the scaling rule: surface is an area, volume is a volume.

Hold onto this

Congruent figures match by rigid motion. Similar figures match up to a uniform scale.
In a right triangle, the square on the hypotenuse equals the sum of the squares on the legs.
Area scales with the square of length; volume scales with the cube.
A circle is the set of points at a fixed distance from a center, and π is the constant in both its circumference and its area.

Formulas

Pythagorean theorem

a^{2} + b^{2} = c^{2}

For a right triangle only. c is the side opposite the right angle.

a, b: Legs
c: Hypotenuse

Area of a triangle

A = \frac{1}{2} bh

Any side may be the base. The height is the perpendicular distance from the opposite vertex to the line of that base.

b: Base length
h: Corresponding height

Circle

C = 2 π r, A = π r^{2}

Circumference grows like the radius; area grows like the radius squared.

r: Radius
π: About 3.14159, exactly the circle constant

Sphere

V = \frac{4}{3} π r^{3}, S = 4 π r^{2}

Volume and surface area. Notice S = dV/dr, a preview of calculus.

V: Volume
S: Surface area

Similar triangles

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}} = k

Corresponding sides stand in one common ratio k. Corresponding angles are equal.

k: Scale factor

Worked examples

1. A ladder against a wall

A 5 m ladder rests against a wall, with its foot 3 m from the wall. How high does it reach?

The wall is perpendicular to the ground, so the triangle is right-angled.
3² + h² = 5².
h² = 25 − 9 = 16.
h = 4, taking the positive length.

4 meters. This is the 3–4–5 triangle.

2. Scale the area, not the length

A circular patio of radius 2 m is replaced by one of radius 3 m. By what factor does the area grow?

Area is πr², so the ratio is (π·3²)/(π·2²) = 9/4.
The linear scale factor is 3/2, and (3/2)² = 9/4.
Numerically, about 12.57 m² becomes about 28.27 m².

The area grows by a factor of 9/4 = 2.25, not by 3/2.

Try these

A triangle has sides 7, 8, and 10. Is the angle opposite 10 a right angle?

Similar triangles have a scale factor of 3. The smaller has area 6. What is the larger area?

Common mistakes

The Pythagorean theorem needs a right angle. On an acute or obtuse triangle it is false; the law of cosines is the replacement.
Height must be perpendicular to the chosen base. A slanted side is not automatically the height.
Doubling the radius doubles the circumference but quadruples the area.