Plane shapes · 3 min read
Rectangle Area: the method, worked through
A rectangle partitions into equal rows and columns of square units.
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Rectangle Area
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Plane shapes01
Rectangle Area
A rectangle partitions into equal rows and columns of square units.
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Rectangle Perimeter
Perimeter measures the full boundary of a rectangle.
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Rectangle Diagonal
The diagonal joins opposite corners and forms a right triangle with the two sides.
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Square Area
A square has two equal perpendicular dimensions.
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Square Perimeter
A square has four equal edges, so one side determines the full boundary.
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Square Diagonal
A square diagonal is the hypotenuse of an isosceles right triangle.
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Circle Area
Circle area depends on the square of the distance from the centre to the boundary.
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Circle Circumference
The circumference is the length around a circle.
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Circle Radius from Area
Inverting the circle-area relation recovers a radius from a known surface area.
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Circle Diameter from Circumference
The ratio of a circle's circumference to its diameter is constant, so division by pi reverses the measurement.
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Triangle Area
A triangle fills half a parallelogram with the same base and perpendicular height.
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Heron Triangle Area
Heron's formula finds triangle area from the three side lengths alone.
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Right Triangle Hypotenuse
The hypotenuse sits opposite the right angle and is always the longest side of a right triangle.
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Missing Right Triangle Leg
Rearranging Pythagoras finds the missing perpendicular leg when the longest side and one leg are known.
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Equilateral Triangle Area
An equilateral triangle divides into two matching right triangles, which explains the square-root-of-three factor.
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Equilateral Triangle Height
The altitude of an equilateral triangle bisects its base and forms a thirty-sixty-ninety triangle.
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Triangle Third Angle
Interior angles of a Euclidean triangle sum to one hundred eighty degrees.
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Law of Cosines Side
The law of cosines extends the right-triangle relation to any included angle.
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Parallelogram Area
Sliding a corner triangle to the other side turns a parallelogram into a rectangle without changing its area.
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Trapezoid Area
A trapezoid's average parallel width multiplied by its perpendicular height gives the enclosed area.
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Rhombus Area
The perpendicular diagonals of a rhombus split it into four right triangles.
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Ellipse Area
An ellipse can be viewed as a circle stretched independently along two perpendicular axes.
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Ellipse Perimeter Estimate
Ramanujan's first perimeter approximation gives a convenient estimate without evaluating an elliptic integral.
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Sector Area
A sector is a fraction of a circle selected by its central angle.
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Arc Length
Arc length measures the curved boundary selected by a central angle.
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Chord Length
A chord is the straight segment between two points on a circle.
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Annulus Area
An annulus is the ring left after removing a concentric smaller disk.
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Regular Polygon Area
A regular polygon splits into congruent triangles around its centre.
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Regular Polygon Interior Angle
A regular polygon shares its total interior angle sum evenly across its vertices.
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Polygon Diagonal Count
Each vertex connects by diagonals to all except itself and its two neighbours.
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Cube Volume
A cube stacks equal square layers through its depth.
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Cube Surface Area
Six identical square faces form the outside of a closed cube.
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Cuboid Volume
Three perpendicular dimensions determine the capacity of a rectangular box.
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Cuboid Surface Area
A cuboid has three pairs of matching rectangular faces.
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Cuboid Space Diagonal
The space diagonal connects opposite vertices through the interior of the box.
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Cylinder Volume
A right circular cylinder has a constant disk-shaped cross-section along its height.
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Cylinder Total Surface Area
Two disks and one unrolled rectangular side make up a closed cylinder's surface.
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Cone Volume
A cone occupies one third of a cylinder with the same base and perpendicular height.
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Cone Slant Height
The side length of a right cone forms a right triangle with its radius and vertical height.
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Cone Total Surface Area
The total combines a circular base with the curved lateral surface.
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Sphere Volume
Sphere volume scales with the cube of radius.
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Sphere Surface Area
The surface of a sphere has four times the area of its great-circle disk.
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Hemisphere Volume
A hemisphere is one half of a sphere cut through its centre.
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Square Pyramid Volume
A square pyramid uses one third of its base-area prism volume.
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Conical Frustum Volume
A frustum describes a cone with its tip removed by a parallel cut.
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Distance Between 2D Points
Euclidean distance combines perpendicular coordinate differences.
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Distance Between 3D Points
Three-dimensional distance extends the same right-angle relation to three orthogonal axes.
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Degrees to Radians
Radians express angle as arc length divided by radius.
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Radians to Degrees
Converting to degrees expresses an angle in three hundred sixty parts per turn.
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Scale Model Length
A one-to-n model reduces every real length by the scale denominator.
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