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What is Geometry?

Perhaps you’re not that into geometric shapes and diagrams because it reminds you a little of the abstract art you pretended to understand, or perhaps you’re wondering if there is more to this subject than another Ed Sheeran song. Whatever the case may be, if you’re reading this you’re on a steady, 45° inclination towards success.

Geometry is the branch of mathematics that focuses on the properties of lines, points, surfaces, and two-dimensional and three-dimensional shapes. This study is important because it tells us the volume, area, length and width, and dimensions of different figures, which comprise of the world around us.

The basics of Geometry

A point is a precise position on a plane, usually represented by a dot. Points have no dimension but are only a precise location.

Lines are straight, extending infinitely in both directions, and have no thickness. Infinite points combined together form a line. A line segment has a starting point and an ending point, while a ray has a starting point but no end point.

Lines - Geometric Shapes

What are the basic shapes in geometry?

Geometric shapes can be either two-dimensional (2D) or three-dimensional (3D). Below, we categorize 2D shapes and their characteristics, and then 3D shapes and their characteristics. The list of characteristics and variety of shapes is by no means exhaustive and is intended to provide a cursory understanding of geometric shapes.

TWO-DIMENSIONAL SHAPES

Triangles

 A Right Triangle Right triangles have one right angle, equal to 90°
 Acute Triangle Acute triangles are those where each angle is below 90°, and all angles are acute
Obtuse Triangle Obtuse triangles are those where at least one angle is greater than 90° or is obtuse and the other two angles are less than 90° and are acute
 Isosceles Triangle Isosceles triangles have at least 2 equal angles and 2 sides of equal length. Hence, all isosceles triangles have a line of symmetry.
Scalene Triangle None of the angles are equal, and none of the none of the sides is the same length in scalene triangles
 An Equilateral Triangle When all three angles are equal to 60°, and all sides are equal in length, it is considered an equilateral triangle

Quadrilaterals

Any polygon (multi-sided shape) that has four sides is called a quadrilateral. Other ways to describe these include quadrangles or tetragons.

 Rectangle A rectangle has 4 sides and 4 right angles. 2 parallel lines are of equal length. If all four parallel lines are of equal length, a rectangle will also be considered a square (although not all rectangles are squares).
 Square Squares have 4 equal sides and 4 right angles. All squares are rectangles, rhombuses, and parallelograms (see below).
 Rhombus A rhombus has 4 equal sides, where opposite sides are parallel, and at least two sides are equal. Rhombuses are parallelograms.
 A ParallelogramParallelogram A parallelogram is 4 sided, where each of the opposite sides is parallel. Sometimes these lines are of equal length, but most of the time they are not.
 A Kite A kite has two pairs of sides of equal length, which are next to each other.
 A Trapezoid A trapezoid US (trapezium UK) has one pair of parallel sides, which may or may not be equal in length.
 Trapezium A trapezium US (trapezoid UK) is a four-sided shape, which has no parallel side.

Sometimes polygons can be convex or concave in shape. In convex polygons, each angle is less than 180°, and concave polygons have one reflex angle that is above 180°. Essentially, concave shapes do not have dents, while convex shapes do. Triangles are always convex.

Examples of concave and convex polygons

Polygon

convex hexagon is a six-sided shape, where each angle is greater than 180° but has no reflex angle.

concave hexagon is a six-sided shape, where at least one reflex angle is greater than 180°.

Circles

A simple, closed shape, a circle consists of a point called the center, and any point on the circle from the center is equidistance. A circle has infinite lines of symmetry.

Circle - Geometrical Shape

Ellipse

An ellipse is a stretched version of a circle. These shapes usually have two lines of symmetry, and the longest diameter of the ellipse is called a major axis, and the shortest diameter is called a minor axis.

3D Geometric Shapes

If your head is reeling from all the possible shapes Ed Sheeran could be singing about, don’t worry –there’s more. Common features for most all 3D shapes is that they consist of some number of faces, edges, and vertices. Below is a list of the basic 3D shapes, which by no means is exhaustive!

Sphere Spheres either have 0 or 1 face, 0 edges and 0 vertices
 Ellipsoids Ellipsoids have either 0 or 1 face, 0 edges, and 0 vertices
 Cylinder Cylinders have 2 or 3 faces, 0 or 2 edges, and 0 vertices
 Cone Cones have 1 or 2 faces, 0 or 1 edges, and 1 apex or vertex
 Cube A cube has 6 faces, 12 edges, and 8 vertices. Each side on the cube are of equal length, and each face is a square. A cube is a type of cuboid (see below).
 Cuboid A cuboid has 6 faces, 12 edges, and 8 vertices, and every face on a cuboid is rectangular in shape.
 Triangular Prism A triangular prism has 5 faces, 9 edges, and 6 vertices. On two ends, the faces are triangular in shape and the rest of the faces are rectangular.
 Hexagonal Prism A hexagonal prism has 8 faces, 18 edges, and 12 vertices. On each end, the faces are hexagonal in shape, while all the other faces are rectangular.
 Triangular Based Pyramids Triangular based pyramids have 4 faces, 6 edges, and 4 vertices. All of the faces are triangular, as is the base. If each triangular face is equilateral then the prism is also called a tetrahedron.
 Square Pyramid Square based pyramids have 5 faces, 8 edges, and 5 vertices. While the base is square shaped, the other 4 faces are triangular.

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