Precision Knowledge Base

Geometry Theorems & Postulates Library

Explore the fundamental laws of Euclidean geometry. Visual proofs, strict definitions, and calculation formulas at your fingertips.

a b c

Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.

$$a^2 + b^2 = c^2$$
A B C

Triangle Sum Theorem

The sum of the interior angles of any triangle in Euclidean geometry is always 180 degrees.

$$\angle A + \angle B + \angle C = 180^\circ$$
r

Area of a Circle

The area of a circle is equal to the product of Pi ($\pi$) and the square of its radius.

$$A = \pi r^2$$

Thales’s Theorem

Any angle inscribed in a semicircle that connects to the diameter ends is a right angle.

$$\text{If } AC \text{ is dia., } \angle ABC = 90^\circ$$
A B C

Law of Sines

The ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
a b c

Law of Cosines

A generalization of the Pythagorean theorem for non-right triangles relating sides and one angle.

$$c^2 = a^2 + b^2 – 2ab \cos C$$

SSS Congruence

If three sides of one triangle are equal to three sides of another, the triangles are congruent.

$$\triangle ABC \cong \triangle DEF$$

SAS Congruence

If two sides and the included angle of one triangle are equal to the corresponding parts of another, they are congruent.

$$\text{Side-Angle-Side}$$

AA Similarity

If two angles of one triangle are equal to two angles of another, the triangles are similar.

$$\triangle ABC \sim \triangle DEF$$
ext

Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles.

$$\angle \text{ext} = \angle A + \angle B$$

Inscribed Angle Theorem

An angle $\theta$ inscribed in a circle is half of the central angle $2\theta$ that subtends the same arc.

$$\angle \text{inscribed} = \frac{1}{2} \angle \text{central}$$
a b c

Heron’s Formula

A method for calculating the area of a triangle when you know the lengths of all three sides.

$$A = \sqrt{s(s-a)(s-b)(s-c)}$$

Isosceles Triangle Thm

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

$$\angle \text{Base}_1 = \angle \text{Base}_2$$
h b

Parallelogram Area

The area of a parallelogram is the product of its base and its vertical height.

$$A = b \cdot h$$
r

Volume of a Sphere

The volume of a sphere depends solely on its radius and is given by a cubic formula.

$$V = \frac{4}{3} \pi r^3$$