Geometry doesn’t have to be complicated and filled with mind-breaking theorems. Mathematics is actually quite simple when you discover the keys that unlock its logical paths. Some things never change and that is also the case in geometry.
We have compiled a list of the most common geometrical subjects and the simple way to solve them. These tricks will actually help you understand geometry better and offer you a more clear perspective on problems.
About Angles…
- If the sum of any two angles is 90°, then the two angles are complementary angles.
- If the sum of any two angles is 180°, then the two angles are supplementary angles.
- The sum of all interior angles of a triangle is 180° and the sum of all exterior angles of a triangle is 360°.
- In a parallelogram, two opposite angles are equal.
About Triangles
– The sum of any two sides of a triangle is greater than the third side. And the difference between any two sides of a triangle is less than the third side.
– The orthocenter, the centroid, and the circumcenter of a triangle are all on a straight line, which is called the Euler line.
– The Triangle of Pascal is a triangular array of numbers where each number on the “interior” of the triangle is the sum of the two numbers directly above.
– A diagonal line cuts a parallelogram in two congruent triangles.
About Squares
- A square has 4 sides of equal length, thus it is also called a regular quadrilateral.
- However, amongst all the other quadrilaterals with the same perimeter, the square has the largest area.
- Different from triangles, the sum of the internal angles of a square is 360°.
- A cube is a 3D square.
- A square is also a rhombus with right angles and a rectangle with equal sides.
- The perimeter of a square is 4 times the length of one side.
About Circles
- Any point on the edge of a circle is at an equal distance from the center of the circle;
- Circle actually refers to the boundary of the shape while the disk is used to refer to the whole shape, including the inside.
- A circle has the shortest perimeter of all shapes with the same area.
- The unit circle is centered at the coordinates (0,0) and can be used to model any trigonometric function.
- When two lines intersect inside a circle, then the measures of the segments of each line multiplied with each other is equal to the product from the other line.
Common Knowledge About Geometry
Geometry is a very old, almost ancient discipline. It was used to build up the pyramids, entire infrastructures of lost civilizations and it’s a part of our daily lives as everything is built on either circles or squares. Hence, it is very important and useful to study geometry, at least at a basic level. In order to understand how fascinating geometry is, take a look at this historical facts:
- The compass is one of the oldest and most used tools in geometry;
- Babylonians discovered the measurement of a circle which was approximately 3 times the diameter. It is very similar to the measurement of Pi (around 3.14).
- Leonardo Da Vinci and Albrecht Dürer have discovered ways to represent 3D objects on 23 plans. Thus, from the 19-20th centuries further research has been done on the subject leading to projective geometry used nowadays in computer graphics.
- The measurement of paper sizes (e.g. A4) use a 1:√2 ratio. If you cut the sizes in half crosswise, the same ratio will be maintained. It’s a great tool for scaling.
- Banach–Tarski paradox states that if a 3D object is decomposed into smaller pieces, those pieces when put together can create two identical copies of the original object. Do you think it’s true?
- Eratosthenes estimated the Earth’s circumference using basic mathematics and he made it with a high precision.
- Apparently, followers of the Pythagora school used small rocks to represent numbers while solving mathematical equations. Calculus means pebbles in Greek. Can you try to solve a mathematical problem using rocks or pebbles?
- Geometry means “geo” – earth and “metria” – measurement, from Greek.
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