Essential Geometry Formulas Cheat Sheet

Grade 7–10 complete reference. Area, volume, Pythagorean theorem, and angle rules.

Geometry is formula-driven. Knowing the right formula instantly converts a hard-looking problem into a straightforward calculation. Below is a complete reference for middle school and high school geometry — organized by topic, with worked examples for each formula.

Area of 2D Shapes

Shape	Formula	Example
Rectangle	A = l × w	l=8, w=5 → A=40
Triangle	A = ½ × b × h	b=10, h=6 → A=30
Circle	A = π × r²	r=5 → A≈78.5
Trapezoid	A = ½(a+b) × h	a=6,b=10,h=4 → A=32
Parallelogram	A = b × h	b=9, h=5 → A=45

Pythagorean Theorem

In any right triangle: a² + b² = c², where c is the hypotenuse.

    Example: a=3, b=4, find c.

    c² = 3² + 4² = 9 + 16 = 25

    c = √25 = 5 ✓

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. If you spot a multiple (like 6-8-10 = 2×(3-4-5)), you already know the answer.

Volume of 3D Solids

Solid	Formula	Key Note
Rectangular Prism (Box)	V = l × w × h	All three dimensions
Cylinder	V = π r² h	r = radius of circle base
Cone	V = ⅓ π r² h	⅓ of cylinder with same r, h
Sphere	V = (4/3) π r³	r = radius of sphere
Pyramid	V = ⅓ × B × h	B = area of base

Angle Relationships

Supplementary angles — two angles that sum to 180°
Complementary angles — two angles that sum to 90°
Vertical angles — opposite angles formed by two intersecting lines; always equal
Triangle angle sum — the three interior angles of any triangle sum to 180°
Exterior angle theorem — an exterior angle equals the sum of the two non-adjacent interior angles
Corresponding angles — when a transversal crosses parallel lines, corresponding angles are equal
Alternate interior angles — also equal when lines are parallel

Perimeter & Circumference

Rectangle perimeter: P = 2(l + w)
Square perimeter: P = 4s
Circle circumference: C = 2πr = πd
Triangle perimeter: P = a + b + c (sum of all three sides)

🎯 Practice with the Geometry Generator

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Frequently Asked Questions

What is the formula for the area of a circle?

Area = π × r², where r is the radius. Using π ≈ 3.14 for school calculations gives A ≈ 3.14 × r².

How do you find the volume of a cylinder?

Volume = π × r² × h, where r is the radius of the circular base and h is the height of the cylinder.

What is the Pythagorean Theorem?

In a right triangle, a² + b² = c², where c is the hypotenuse. Common triples: 3-4-5, 5-12-13, 8-15-17.

What do supplementary angles add up to?

Supplementary angles add up to 180°. Complementary angles add up to 90°. Vertical angles are always equal to each other.

What are the angles in a triangle?

The three interior angles of any triangle always sum to 180°. This is the Triangle Angle Sum theorem.