Math explainers · 13 pages
Geometry explainers
Clear, citable explainers for the foundational concepts of geometry — shapes, angles, distance, midpoint, area of circles/triangles, perimeter, and surface area.
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All 13 geometry explainers pages
Distance formula
The distance formula calculates the straight-line distance between two coordinates, (x₁, y₁) and (x₂, y₂), in a Cartesian plane. It is expressed as d = √((x₂ − x₁)² + (y₂ − y₁)²). Derived from the Pythagorean theorem, it determines the length of the hypotenuse by squaring the differences in the coordinates, summing them, and taking the principal square root.
DefinitionAngles
An angle is a measure of rotation formed by two rays, or sides, sharing a common endpoint called the vertex. It is measured in degrees (°) or radians (rad). Angles are classified into distinct types based on their size: acute, right, obtuse, straight, and reflex, spanning from 0° to a full 360° rotation.
DefinitionArea of a circle
The area of a circle is the measure of the flat space enclosed by its circular boundary. The formula is A = πr², where r is the radius (the distance from the centre to the boundary) and π (pi) is a mathematical constant approximately equal to 3.14159. If using the diameter, d, it is A = π(d/2)².
DefinitionMidpoint formula
The midpoint formula calculates the exact centre point of a line segment connecting two coordinates, (x₁, y₁) and (x₂, y₂). The formula is M = ((x₁ + x₂)/2, (y₁ + y₂)/2). It works by separately calculating the arithmetic mean of the x-coordinates and the y-coordinates of the two endpoints.
DefinitionEllipse
An ellipse is a regular oval shape defined by two internal focal points (foci). For any point on the boundary, the sum of its distances to the two foci is constant. Its standard Cartesian equation is (x²/a²) + (y²/b²) = 1, where a and b are the lengths of the semi-major and semi-minor axes.
DefinitionPerimeter
The perimeter is the path length that outlines a two-dimensional shape. For polygons, it is calculated by adding the lengths of all the outer sides. For curved shapes like a circle, the perimeter is called the circumference and is calculated using the radius or diameter.
DefinitionArea of a triangle
The area of a triangle measures the region enclosed within its three vertices. The standard formula is A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex. Alternatively, if only the side lengths are known, Heron’s formula can be used.
DefinitionSurface area
Surface area is the sum of the areas of all the exterior surfaces of a three-dimensional object. It is divided into lateral surface area (the area of the sides) and total surface area (which includes the bases). It is measured in square units, such as square metres or square centimetres.
DefinitionTypes of triangles
Triangles are classified using two main criteria: side lengths and interior angles. By sides, they are equilateral (three equal sides), isosceles (two equal sides) or scalene (no equal sides). By angles, they are acute (all angles < 90°), right (one angle = 90°) or obtuse (one angle > 90°).
DefinitionGeometry
Geometry is the study of shape, size, relative position of figures, and the properties of space. The word is derived from the Greek "geometria," meaning "earth-measuring." It is split into plane geometry (flat 2D shapes) and solid geometry (3D shapes). It forms the basis of physics, architecture, engineering, and astronomy.
DefinitionPythagorean theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle, c) is equal to the sum of the squares of the other two sides (legs a and b). The algebraic formula is a² + b² = c². It is one of the most famous theorems in mathematics and is used to calculate distances and solve triangles.
DefinitionArea
Area is a quantity that expresses the extent of a two-dimensional figure in the plane. It measures the amount of space inside a shape. Standard formulas are used to calculate the area of regular shapes (such as rectangles, triangles, and circles), while irregular shapes require decomposition or integration. It is measured in square units (e.g. square metres, cm²).
DefinitionCircumference
Circumference is the term used for the perimeter of a circle or ellipse. The formula to calculate the circumference of a circle is C = 2πr or C = πd, where r is the radius, d is the diameter, and π (pi) is a mathematical constant approximately equal to 3.14159. It is measured in linear units such as metres or centimetres.
