Free AI Geometry Solver

Geometry combines visual reasoning with algebraic calculation in ways that many students find challenging. Our free AI geometry solver interprets diagrams, applies theorems, and produces complete step-by-step solutions for every type of geometry problem you encounter in US high school and college courses.

Triangle Geometry

Triangles are the most heavily tested shape in geometry courses. The AI solver handles triangle congruence proofs using SSS, SAS, ASA, AAS, and HL criteria. It applies the Pythagorean theorem and its converse, calculates areas using multiple formulas (base-height, Heron's formula, coordinate method), and solves problems involving similar triangles and proportional reasoning.

For right triangles, the solver applies trigonometric ratios (sine, cosine, tangent) and their inverses to find missing sides and angles. It handles special right triangles (30-60-90 and 45-45-90) and applies the Law of Sines and Law of Cosines for non-right triangles.

Circle Geometry

Circle problems require knowledge of many specific theorems about chords, tangents, secants, arcs, and inscribed angles. The AI applies the inscribed angle theorem, central angle theorem, tangent-radius perpendicularity, chord-chord power of a point, and secant-tangent relationships. It calculates arc lengths, sector areas, and segment areas with clear formula application.

Equation of a circle problems in both standard and general form are solved with conversion steps shown. The solver identifies center and radius from any form and handles problems involving tangent lines to circles.

Geometric Proofs

Two-column proofs are one of the most difficult topics for geometry students. The AI solver constructs proofs by identifying given information, determining what needs to be proved, selecting appropriate theorems and postulates, and organizing the reasoning into a logical sequence. Each statement is paired with a justification.

The solver handles proofs involving parallel lines and transversals, triangle congruence and similarity, quadrilateral properties, and circle theorems. It can also construct paragraph proofs and coordinate geometry proofs when those formats are required.

Quadrilaterals and Polygons

Properties of parallelograms, rectangles, rhombi, squares, trapezoids, and kites are applied to solve for missing measurements and prove relationships. The solver also handles regular polygon problems including interior and exterior angle sums, apothem calculations, and area formulas.

Coordinate Geometry

The solver calculates distances, midpoints, slopes, and equations of lines in coordinate geometry problems. It proves geometric properties using coordinates, finds areas of polygons using the shoelace formula, and determines whether points are collinear. Transformation problems including translations, reflections, rotations, and dilations are solved with coordinate rules clearly stated.

Surface Area and Volume

Three-dimensional geometry problems involving prisms, cylinders, pyramids, cones, and spheres are solved with surface area and volume formulas applied step by step. Composite solid problems that combine multiple shapes are broken down into individual components. Cross-section analysis and solid of revolution problems bridge geometry with calculus.

Circles in the Coordinate Plane

The solver handles the equation of a circle in standard form and general form, converting between the two using completing the square. It finds the center and radius from any form, determines whether a point lies inside, on, or outside a circle, finds the equations of tangent lines at given points, and solves problems involving intersecting circles and the radical axis. These coordinate geometry circle problems are a staple of both the SAT and high school geometry finals. The solver also demonstrates how to write the equation of a circle given three points on its circumference, a technique that combines systems of equations from algebra with geometric understanding.

Trigonometry in Geometry

Trigonometry bridges geometry and algebra, and our solver handles both the geometric and algebraic aspects. Right triangle trigonometry using SOH-CAH-TOA, the unit circle, radian and degree conversion, trigonometric identities, and solving trigonometric equations are all supported. The solver applies the Law of Sines and Law of Cosines to non-right triangles, calculates areas using the formula A = (1/2)ab sin(C), and solves application problems involving angles of elevation and depression. These skills are tested heavily on the SAT and in pre-calculus courses.

Constructions and Transformations

Geometric constructions using compass and straightedge are explained with step-by-step procedures. The solver describes how to bisect angles, construct perpendicular lines, inscribe polygons in circles, and create parallel lines through given points. Transformations including translations, reflections, rotations, and dilations are solved with coordinate rules and composition sequences. The solver identifies lines of symmetry, centers of rotation, and scale factors for similarity transformations.

Three-Dimensional Geometry and Cross-Sections

Beyond surface area and volume, the solver handles cross-section analysis, identifying the shapes produced when a plane intersects a solid at various angles. It calculates the dimensions of cross-sections, determines loci of points in three-dimensional space, and solves problems involving distances between skew lines and planes. For students preparing for the AP Calculus exam, these three-dimensional visualization skills provide essential preparation for solids of revolution and related volume calculations.

Geometry on Standardized Tests

Geometry and trigonometry account for approximately 15% of SAT math questions. These questions frequently combine geometric reasoning with algebraic problem-solving, requiring students to set up equations based on geometric relationships and then solve them. Our solver helps you practice this integration of skills by showing both the geometric setup and the algebraic solution. For word problems involving geometric contexts like area optimization or distance calculations, the solver translates the verbal description into a diagram-based approach.

Geometry builds spatial reasoning skills that are essential for physics, engineering, and architecture. Use our AI math solver to master these concepts at your own pace. Compare our solver with other tools on our best AI math solver page, or explore our dedicated statistics solver for data analysis help.