What are the basic geometry formulas?

The most essential geometry formulas are: rectangle area = l × w; triangle area = ½ × b × h; circle area = πr² and circumference = 2πr; Pythagorean theorem c² = a² + b²; and the interior angle sum of a triangle = 180°. These five cover the majority of high school geometry problems.

What is the formula for the area of a triangle?

The area of a triangle is A = ½ × b × h, where b is the base and h is the perpendicular height from the base to the opposite vertex. The base and height must be perpendicular to each other — do not use the slant side as the height.

How do you find the volume of a cylinder?

The volume of a cylinder is V = πr²h, where r is the radius of the circular base and h is the height of the cylinder. Multiply the area of the circular base (πr²) by the height. For example, a cylinder with radius 3 and height 5 has volume = π × 9 × 5 = 45π ≈ 141.4 cubic units.

What is the Pythagorean theorem formula?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b², where c is the hypotenuse (the side opposite the right angle) and a and b are the two legs.

Geometry Formulas Cheat Sheet: Every Formula for 2D and 3D Shapes

This geometry formulas cheat sheet is a single printable reference covering every formula you need for high school geometry: area and perimeter for five 2D shapes, volume and surface area for six 3D solids, the Pythagorean theorem, and angle relationship formulas — including the polygon interior angle formula that most competitors skip entirely.

The page is print-optimized (Cmd+P / Ctrl+P). Each formula links to a free step-by-step calculator so you can verify any answer. Formulas align with standard US high school geometry curricula. Last reviewed September 2026.

How to Use This Geometry Formula Sheet

Use the Quick-Reference Table below to scan all formulas at once.
Jump to the 2D Shapes section for area and perimeter formulas with variable definitions.
Jump to the 3D Shapes section for volume and surface area formulas.
Use the Pythagorean Theorem and Angle Relationships sections for the formulas most often missed on cheat sheets.
Click any calculator link to verify a formula with step-by-step working.

Quick-Reference Summary Table

Shape	Area / Volume	Perimeter / Surface Area
Rectangle	A = l × w	P = 2(l + w)
Triangle	A = ½ × b × h	P = a + b + c
Circle	A = πr²	C = 2πr
Trapezoid	A = ½(b₁ + b₂) × h	P = b₁ + b₂ + leg₁ + leg₂
Parallelogram	A = b × h	P = 2(b + s)
Rectangular prism	V = l × w × h	SA = 2(lw + lh + wh)
Cube	V = s³	SA = 6s²
Cylinder	V = πr²h	Total SA = 2πr² + 2πrh
Sphere	V = (4/3)πr³	SA = 4πr²
Cone	V = (1/3)πr²h	Total SA = πr² + πrl
Triangular prism	V = ½ × b × h × l	SA = bh + (a + b + c) × l
Pythagorean theorem	c² = a² + b²	—

2D Shape Formulas (Area and Perimeter)

These 2D geometry formulas cover the five shapes that appear most often in high school geometry courses. For each shape, h always means the perpendicular height — not a slant side.

Rectangle

Area = l × w

Perimeter = 2(l + w)

l = length, w = width. A square is a rectangle where l = w, so Area = s² and Perimeter = 4s.

Example: Rectangle with l = 8 and w = 5 → Area = 40, Perimeter = 26

Triangle

Area = ½ × b × h

Perimeter = a + b + c

b = base, h = perpendicular height from the base to the opposite vertex, a, b, c = the three side lengths.

Need to find a missing side of a right triangle? Use our Pythagorean theorem calculator — enter any two sides to find the third.

Circle

Area = πr²

Circumference = 2πr

Arc length = (θ/360°) × 2πr

Sector area = (θ/360°) × πr²

r = radius (half the diameter), θ = central angle in degrees. Arc length and sector area formulas use the angle to find a portion of the full circle.

Common mistake: Using diameter (d) where the formula expects radius (r). Always check: r = d/2. The area formula is πr², not π(2r)² or πd².

Find circumference, arc length, and sector area with the circle calculator — enter any one measurement to get all the others.

Trapezoid

Area = ½(b₁ + b₂) × h

Perimeter = b₁ + b₂ + leg₁ + leg₂

b₁ and b₂ are the two parallel sides (bases). h = perpendicular distance between the two parallel sides. leg₁ and leg₂ = the non-parallel sides.

Important: b₁ and b₂ must be the parallel sides, not any two sides. The height h is the perpendicular distance between them, not the length of a leg.

Parallelogram

Area = b × h

Perimeter = 2(b + s)

b = base, h = perpendicular height (not the slant side s), s = slant side length.

Common mistake: Using the slant side length as h. The height must be perpendicular to the base — it is the vertical distance, not the diagonal side.

Calculate area and perimeter for any of these shapes with the area and perimeter calculator — includes step-by-step working and diagrams.

3D Shape Formulas (Volume and Surface Area)

These 3D geometry formulas cover the six solids most commonly tested in high school geometry. Volume is measured in cubic units; surface area in square units.

Rectangular Prism (Box)

Volume = l × w × h

Surface Area = 2(lw + lh + wh)

l = length, w = width, h = height. The surface area adds up the areas of all six rectangular faces (three pairs of opposite faces).

Cube

Volume = s³

Surface Area = 6s²

s = side length. A cube has 6 identical square faces; total SA = 6 × s².

Cylinder

Volume = πr²h

Lateral Surface Area = 2πrh

Total Surface Area = 2πr² + 2πrh

r = radius of the circular base, h = height. Lateral SA is the curved side only; total SA adds the two circular bases (2 × πr²).

Sphere

Volume = (4/3)πr³

Surface Area = 4πr²

r = radius.

Common mistake: Using diameter instead of radius. The sphere formulas use r (radius), not d (diameter). If you are given the diameter, divide by 2 before substituting.

Cone

Volume = (1/3)πr²h

Lateral Surface Area = πrl

Total Surface Area = πr² + πrl

r = radius of the base, h = perpendicular height, l = slant height.

Finding slant height: l = √(r² + h²) — use the Pythagorean theorem on r and h. If the problem gives the slant height directly, use it as l.

Triangular Prism

Volume = ½ × b × h × l

Surface Area = bh + (a + b + c) × l

b = base of the triangular face, h = height of the triangular face, l = length of the prism, a, b, c = sides of the triangle.

Check your volume and surface area answers with the volume and surface area calculator — covers all six 3D shapes above.

Pythagorean Theorem

c² = a² + b²

c = hypotenuse (the side opposite the right angle, always the longest side), a and b = the two legs. Use this formula to find a missing side in any right triangle.

Worked example (3-4-5 triangle):

Known: a = 3, b = 4, find c
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5

Finding a leg (solving for a): a² = c² − b² → a = √(c² − b²)

Use the Pythagorean theorem when: you have a right triangle (one 90° angle) and know two of the three side lengths. It does not apply to non-right triangles — use the law of cosines for those.

Solve for any missing side step-by-step with the Pythagorean theorem calculator — includes a labeled diagram.

Angle Relationship Formulas

These angle formulas are completely absent from most geometry cheat sheets — yet they appear constantly on geometry tests and the SAT/ACT.

Relationship	Formula / Rule
Complementary angles	∠A + ∠B = 90°
Supplementary angles	∠A + ∠B = 180°
Vertical angles	∠A = ∠B (opposite angles when two lines cross)
Triangle interior angles	∠A + ∠B + ∠C = 180°
Polygon interior angle sum	Sum = (n − 2) × 180°
Interior angle of regular polygon	Each angle = (n − 2) × 180° / n
Exterior angle of any regular polygon	Each exterior angle = 360° / n
Exterior angle of a triangle	= sum of the two non-adjacent interior angles

n = number of sides in the polygon.

Examples using the polygon interior angle formula (n − 2) × 180°:

Triangle (n = 3): (3−2) × 180° = 180°
Quadrilateral (n = 4): (4−2) × 180° = 360°
Pentagon (n = 5): (5−2) × 180° = 540°
Hexagon (n = 6): (6−2) × 180° = 720°
Octagon (n = 8): (8−2) × 180° = 1,080°

Common Geometry Formula Mistakes to Avoid

Using diameter instead of radius in circle formulas: Area = πr² uses r. If given diameter d, compute r = d/2 first.
Using the slant height instead of perpendicular height in cone volume: V = (1/3)πr²h uses the vertical height h, not the slant height l.
Forgetting to square r in πr²: Area = π × r² — the radius is squared, not multiplied by π without squaring.
Mixing up b₁ and b₂ in the trapezoid formula: The two parallel sides are b₁ and b₂. The legs are not included in the area formula, only in the perimeter formula.
Using slant side as height in a parallelogram: The height h in A = b × h must be perpendicular to the base — it is never the diagonal or slant side.
Applying the Pythagorean theorem to non-right triangles: c² = a² + b² only works when one angle is exactly 90°. For other triangles, use the law of cosines.

Frequently Asked Questions

What are the basic geometry formulas?

The most essential geometry formulas are: rectangle area = l × w; triangle area = ½ × b × h; circle area = πr² and circumference = 2πr; Pythagorean theorem c² = a² + b²; and the interior angle sum of a triangle = 180°. These five cover the majority of high school geometry problems.
What is the formula for the area of a triangle?

The area of a triangle is A = ½ × b × h, where b is the base and h is the perpendicular height from the base to the opposite vertex. The base and height must be perpendicular — do not use the slant side as the height.
How do you find the volume of a cylinder?

The volume of a cylinder is V = πr²h, where r is the radius of the circular base and h is the height. Multiply the area of the base (πr²) by the height. For example, a cylinder with radius 3 and height 5 has volume = π × 9 × 5 = 45π ≈ 141.4 cubic units.
What is the Pythagorean theorem formula?

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b², where c is the hypotenuse and a and b are the two legs. Use it whenever you know two sides of a right triangle and need to find the third.

This page is print-optimized — press Cmd+P (Mac) or Ctrl+P (Windows) to print or save as PDF. Explore all 27 free math calculators at MathInSite — no sign-up required.

Working with triangles that have no right angle? See Law of Sines vs. Law of Cosines for which rule to use.

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