Geometry Formulas: Calculators, Proofs & Examples

Geometry splits into three areas. Coordinate geometry formulas work on the Cartesian plane: the distance formula, the midpoint formula, and slope all live here. Plane geometry formulas cover flat shapes: triangles, circles, quadrilaterals, and polygons. Solid geometry formulas handle three-dimensional figures: prisms, cylinders, cones, spheres, and pyramids. The calculators on this site are organized by area, starting with coordinate geometry because it appears most often on the SAT, ACT, and high-school exams.

Don't try to memorize all geometry formulas at once. Pick the topic you're currently studying, use the calculator to check your answer, and read the proof section when you want the result to stick long-term. A geometry formula sheet is useful in class, but understanding the proof is what makes the formula memorable on a test.

This is a working list of all geometry formulas on this site. Each page follows the same structure: the formula itself (with clear notation), a free calculator that shows every step, a plain-language explanation of why the formula works, worked examples with real numbers, and practice problems with answers. That's the full picture: from the definition to the derivation to the drill.

Basic geometry formulas

If you're just starting out, these basic geometry formulas are the ones to learn first. Each has a dedicated calculator and proof page on this site.

• Area of a rectangle: A = length x width - the simplest area formula and the foundation for understanding area in general.
• Area of a triangle: A = (1/2) x base x height - works for any triangle. The 1/2 comes from the fact that a triangle is half of a parallelogram.
• Circumference of a circle: C = 2πr - the distance around a circle, where r is the radius.
• Pythagorean theorem: a² + b² = c² - for right triangles, where c is the hypotenuse. This is the backbone of the distance formula.
• Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²] - find the straight-line length between any two coordinate points. The most-tested formula in coordinate geometry.
• Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) - the coordinates of the point exactly halfway between two points on a plane.

Start with the distance formula if you're working on coordinate geometry. It connects directly to the Pythagorean theorem, so once you understand one, you understand both.

Upcoming formula pages

The following calculators and formula pages are in development. Each will include a free calculator, worked examples, a proof, and a FAQ. They'll be linked here once each page is complete and reviewed.

• Midpoint formula: find the point exactly halfway between two coordinates on a plane
• Slope formula: calculate the steepness and direction of a line through two points
• Triangle area formula: area from base and height, or from three coordinate vertices using the coordinate method
• Circle formulas: circumference, area, arc length, and sector area
• Pythagorean theorem: find the missing side of a right triangle; the backbone of the distance formula
• Volume formulas: cylinders, spheres, cones, and prisms
• Surface area formulas: lateral and total surface area for 3D solids
• Area and perimeter formulas: rectangles, triangles, parallelograms, trapezoids
• Circle equation: the standard form of the equation of a circle, built directly from the distance formula

If you need a specific formula for an upcoming test, use the search bar at the top of the page, or check the distance formula page for SAT-specific tips on coordinate geometry. This list of geometry formulas will grow as each new calculator and proof page is published.

Study by Exam

The most-tested geometry formulas vary by exam. Here's a quick guide to what each test emphasizes:

• SAT Math: distance formula, slope, midpoint, Pythagorean theorem, circle area and circumference, right triangle trigonometry
• ACT Math: same core set plus coordinate geometry applied to lines and parabolas
• Common Core (High School Geometry): triangle congruence, coordinate proofs using the distance and midpoint formulas, circle theorems
• GCSE Maths: Pythagoras' theorem, area and perimeter of standard shapes, volume of prisms and cylinders

For all of these exams, the distance formula page is a good starting point. It covers the underlying reasoning that connects the Pythagorean theorem to coordinate geometry, the single biggest geometry concept on every standardized test.

Frequently Asked Questions