Geometry
Free lessons and practice worksheets for geometry — worked examples, common mistakes, and step-by-step solutions.
- 01Geometry3 min read
3D Formulas (Volume & Surface Area)
Three-dimensional formulas calculate volume and surface area for geometric solids like cubes, cylinders, spheres, and cones. Volume measures the space inside a shape, expressed in cubic units, while surface area measures the total area of all exterior faces, expressed in square units. These formulas apply fundamental geometric principles to quantify physical properties of solid objects.
- 02Geometry3 min read
3D Shapes
Three-dimensional shapes are geometric figures that extend in three directions and occupy space in the physical world. These solid objects have measurable properties including faces (flat or curved surfaces), edges (where surfaces meet), and vertices (corner points). Common 3D shapes include cubes with 6 square faces, spheres with 1 curved surface, and cylinders with 2 circular faces connected by 1 curved surface.
- 03Geometry3 min read
Angles
An angle measures the amount of rotation between two rays that share a common endpoint called a vertex. Angles are measured in degrees (°), with a full rotation equaling 360°. The study of angles forms a fundamental component of geometry, appearing in CCSS standards from grade 4 through high school.
- 04Geometry3 min read
Area & Perimeter
Area quantifies the two-dimensional space enclosed within a shape's boundaries, measured in square units like square inches or square feet. Perimeter represents the total distance around a shape's outer edge, measured in linear units like inches or feet. These fundamental geometric measurements apply to rectangles, triangles, circles, and composite shapes.
- 05Geometry3 min read
Circles
A circle is a two-dimensional shape consisting of all points that are equidistant from a central point. The distance from the center to any point on the circle is called the radius, while the distance across the circle through the center is the diameter. Circles appear in CCSS.7.G standards where students learn to calculate circumference using C = 2πr and area using A = πr².
- 06Geometry3 min read
Classify Triangles & Quadrilaterals
Classifying triangles and quadrilaterals involves sorting these shapes based on their side lengths, angle measures, and parallel sides. A triangle with sides 6, 8, and 10 is classified as a scalene right triangle because all sides differ and one angle equals 90°. This classification system helps identify specific properties and relationships within geometric shapes.
- 07Geometry3 min read
Coordinates (First Quadrant)
Coordinates in the first quadrant represent points on a grid where both x and y values are positive numbers. Each point is written as an ordered pair (x, y), where x represents the horizontal distance from the origin and y represents the vertical distance. The first quadrant contains all points where x ≥ 0 and y ≥ 0, making it the foundation for coordinate geometry in elementary mathematics.
- 08Geometry3 min read
Coordinates (Four Quadrants)
A coordinate system with four quadrants divides the plane into regions based on positive and negative values. Quadrant I contains points where both x and y are positive (+, +), Quadrant II has negative x and positive y (−, +), Quadrant III has both coordinates negative (−, −), and Quadrant IV has positive x and negative y (+, −). The quadrants are numbered counterclockwise starting from the upper right.
- 09Geometry3 min read
Geometric Constructions
Geometric constructions involve creating precise figures using only a compass and unmarked straightedge, following rules established by ancient Greek mathematicians. These methods produce exact geometric relationships without requiring numerical measurements or marked rulers. The compass draws circles and arcs, while the straightedge creates straight lines between points.
- 010Geometry3 min read
Polygon Properties
A polygon is a closed figure formed by three or more straight line segments called sides, which meet at points called vertices. Each polygon is classified by its number of sides: a triangle has 3 sides, a quadrilateral has 4 sides, a pentagon has 5 sides, and so on. Regular polygons have all sides equal in length and all angles equal in measure.
- 011Geometry3 min read
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two legs equals the square of the hypotenuse: a² + b² = c². This fundamental geometric relationship, named after ancient Greek mathematician Pythagoras, provides a method to calculate any missing side of a right triangle when two sides are known. The theorem applies specifically to right triangles, which contain one 90-degree angle.
- 012Geometry3 min read
Recognising 2D Shapes
Recognizing 2D shapes involves identifying flat geometric figures by counting their sides, corners, and examining their properties. A triangle has 3 sides and 3 angles that sum to 180°, while a square has 4 equal sides and 4 right angles. These fundamental shapes appear throughout mathematics, from basic geometry in CCSS.2.G standards to advanced calculations involving interior angles.
- 013Geometry3 min read
Shape Properties
Shape properties define the distinctive characteristics of geometric figures, including the types of lines, angles, and symmetries they contain. A rectangle, for instance, has 4 right angles and 2 pairs of parallel sides, while a rhombus has 4 equal sides and 2 pairs of parallel sides but no right angles. These properties serve as identification markers that distinguish one shape from another.
- 014Geometry3 min read
Similarity & Scale Factors
Similar shapes have identical angles but proportional sides, with the scale factor representing the constant ratio between corresponding measurements. A scale factor of 3 means each side of the larger shape is exactly 3 times longer than the corresponding side of the smaller shape. This relationship extends to areas, which scale by the square of the linear scale factor.
- 015Geometry3 min read
Symmetry
Symmetry occurs when a shape can be divided into identical parts that mirror each other exactly. A line of symmetry acts as a fold line that creates two matching halves, while rotational symmetry means a shape looks the same after being turned less than a full 360-degree rotation. These properties appear in regular polygons, where an equilateral triangle has 3 lines of symmetry and a square has 4.
- 016Geometry3 min read
Volume
Volume quantifies the three-dimensional space enclosed within a solid object, measured in cubic units such as cubic centimeters (cm³) or cubic meters (m³). The calculation method depends on the shape: a cube with 4 cm sides has volume 64 cm³, while a rectangular box measuring 6 × 7 × 7 cm contains 294 cm³. Volume formulas multiply length, width, and height dimensions together, with variations for curved shapes like cylinders and spheres.