Geometry Formulas

Geometry is a branch of mathematics that deals with the study of shapes, sizes, positions, and the properties of space. It is a fundamental subject that is used in many fields such as architecture, engineering, physics, and computer graphics. 

• Geometry formulas are essential tools that enable mathematicians and students to solve problems involving lines, angles, triangles, circles, and other shapes. 
• These formulas provide a concise and systematic way to express the relationships between various geometric entities and enable us to calculate their measurements accurately. 
• In this way, they are the foundation upon which geometry rests and are indispensable for understanding the principles of the subject.

Key Terms: Geometry, Square, Rectangle, Cube, Cuboid, Triangle, Area, Perimeter, Height, Base, Quadrilateral, Shapes

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Basic Geometric Shapes and Formulas

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Geometry is all about shapes and their properties. In class 9, students learn about basic geometric shapes such as points, lines, angles, and planes. They also learn about triangles, quadrilaterals, and circles. 

• Square

A square is a four-sided polygon with all sides equal in length and all angles equal to 90 degrees. The area of a square is calculated by multiplying the length of one side by itself, and the perimeter is calculated by adding the length of all four sides. The diagonals of a square are lines that connect opposite vertices. The length of a diagonal can be calculated using the Pythagorean theorem.

Square

• Area: A = s² 
• Perimeter: P = 4s 
• Diagonal: d = s√2, where s is the length of one side

• Rectangle

A rectangle is a four-sided polygon with opposite sides equal in length and all angles equal to 90 degrees. The area of a rectangle is calculated by multiplying the length and width, and the perimeter is calculated by adding the length of all four sides. The diagonals of a rectangle are the lines that connect opposite vertices. The length of a diagonal can be calculated using the Pythagorean theorem.

Rectangle

• Area: A = l x w 
• Perimeter: P = 2l + 2w 
• Diagonal: d = √(l² + w²), where l is the length and w is the width

• Triangle

A triangle is a three-sided polygon. The area of a triangle is calculated by multiplying the base and height and dividing the result by two, and the perimeter is calculated by adding the length of all three sides.

Triangle

• Area: A = 1/2bh (where b is the base and h is the height)
• Perimeter: P = a + b + c (where a, b, and c are the lengths of the three sides)

• Circle

A circle is a two-dimensional shape consisting of points that are equidistant from a center point. The area of a circle is calculated by multiplying pi (π) by the radius squared, and the circumference (perimeter) is calculated by multiplying the diameter by pi.

Circle

• Area: A = πr² (where r is the radius)
• Circumference: C = 2πr (where r is the radius) or C = πd (where d is the diameter)
• Diameter = 2r 

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Geometry Formulas for Quadrilaterals

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A quadrilateral is a geometric shape that has four sides and four angles. The sum of the interior angles of a quadrilateral is always 360 degrees.

• Parallelogram

A parallelogram is a four-sided polygon with opposite sides parallel to each other. The area of a parallelogram is calculated by multiplying the base and height, and the perimeter is calculated by adding the length of all four sides. The diagonals of a parallelogram can be calculated using the Pythagorean theorem.

Parallelogram

• Area: A = bh (where b is the base and h is the height)
• Perimeter: P = 2(a + b) (where a and b are the lengths of adjacent sides)
• Diagonal of a Parallelogram: d = √(a² + b² + 2abcos(θ)) (where a and b are the lengths of adjacent sides and θ is the angle between them)

• Trapezoid

A trapezoid is a quadrilateral with one pair of parallel sides. The area of a trapezoid is calculated by multiplying the average of the parallel sides (also known as the median) by the height, and the perimeter is calculated by adding the length of all four sides.

Trapezoid

• Area: A = (a + b)h/2 (where a and b are the lengths of the parallel sides and h is the height)
• Perimeter: P = a + b + c + d (where a and b are the lengths of the parallel sides, and c and d are the lengths of the non-parallel sides)

• Rhombus

A rhombus is a quadrilateral with all sides equal in length. The area of a rhombus is calculated by multiplying the length of one diagonal by the length of the other diagonal, and dividing the result by two. The perimeter is calculated by adding the length of all four sides. The diagonals of a rhombus are perpendicular bisectors of each other, meaning they intersect at a 90-degree angle and bisect each other. This means that each diagonal divides the rhombus into two congruent right triangles, which can be used to find the length of the diagonals.

Rhombus

• Area: A = (d₁ x d₂)/2 (where d₁ and d₂ are the lengths of the diagonals)
• Perimeter: P = 4s (where s is the length of one side)
• Length of Diagonals: d₁ = 2asin(θ/2) and d₂ = 2acos(θ/2) (where a is the length of one side and θ is the measure of one angle)

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Right Triangles and Pythagorean Theorem

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Right triangles are triangles with one angle measuring 90 degrees, also known as a right angle. They have some unique properties that make them important in mathematics and real-world applications. One of the most fundamental and useful tools for working with right triangles is the Pythagorean theorem.

Right Triangles

Similar to triangle, the area of a right triangle is calculated by multiplying the base and height and dividing the result by two, and the perimeter is calculated by adding the length of all three sides.

• Area: A = ½ bh (where b is the base and h is the height)
• Perimeter: P = a + b + c (where a, b, and c are the lengths of the three sides)

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be written as an equation:

a² + b² = c²

where a and b are the lengths of the two legs (the sides adjacent to the right angle) and c is the length of the hypotenuse.

This theorem can be used to find the length of any side of a right triangle, as long as the lengths of the other two sides are known. It can also be used to determine if a triangle is a right triangle, by checking if the lengths of its sides satisfy the equation a² + b² = c²

Heron's formula

Heron's formula is a formula used to calculate the area of a triangle when the lengths of all three sides are known. The formula is named after the ancient Greek mathematician Hero of Alexandria. Here's the formula:

Let a, b, and c be the lengths of the three sides of a triangle. Then the area of the triangle is given by:

A = √[s(s-a)(s-b)(s-c)]

where s is the semi-perimeter of the triangle, which is half the perimeter, or:

s = (a + b + c)/2

Heron's formula is useful when you know the lengths of all three sides of a triangle but cannot determine its height, or when other methods of calculating the area are too complicated. It is commonly used in geometry and can be applied to any type of triangle, whether it is equilateral, isosceles, or scalene.

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Arc and Sector of a Circle Formula

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A circle is a two-dimensional shape that is defined as a set of points that are equidistant from a fixed point called the center of the circle.

Arc and Sector of a Circle Formula

• Arc of a Circle

An arc of a circle is a portion of the circumference of the circle. The length of an arc can be calculated using the following formula:

Arc Length = (angle θ/360°) x 2πr

where θ is the central angle of the arc (measured in degrees), r is the radius of the circle, and 2πr is the circumference of the circle.

• Sector of a Circle

A sector of a circle is a region bounded by two radii and an arc. The area of a sector can be calculated using the following formula:

Sector Area = (angle θ/360°) x πr²

where θ is the central angle of the sector (measured in degrees), r is the radius of the circle, and πr² is the area of the circle.

Additionally, the length of the arc that forms the sector can be found using the arc length formula mentioned above.

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Congruence and Similarity

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Congruence and similarity are two important concepts in geometry that describe the relationship between different geometric shapes.

Congruence refers to the property of having the same shape and size. Two geometric shapes are said to be congruent if they have the same size and shape, meaning that all corresponding angles and sides are equal. Congruent shapes can be moved, rotated, or flipped without changing their size or shape. Congruence is denoted by the symbol "≅" (congruent sign).

Congruent Triangles

To determine whether two shapes are congruent or similar, one can use various methods such as the angle-angle (AA), side-angle-side (SAS), and side-side-side (SSS) criteria. These criteria are based on the corresponding angles and sides of the two shapes and can be used to show that the shapes are either congruent or similar.

On the other hand, similarity refers to the property of having the same shape, but not necessarily the same size. Two geometric shapes are said to be similar if they have the same shape, meaning that all corresponding angles are equal, but their corresponding sides may be different lengths. Similar shapes can be scaled up or down uniformly, while still maintaining their shape. The similarity is denoted by the symbol "∼" (tilde sign).

Similarity

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Geometry Formulas of Three-Dimensional Shapes

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Three-dimensional shapes, also known as 3D shapes or solids, are objects that have length, width, and height or depth. 

• Cube

A cube is a three-dimensional shape with six square faces of equal size.

Cube

Its formulas are:

• Volume: V = s³
• Surface area: A = 6s²
• Diagonal: d = s√3, where s is the length of any edge of the cube

• Cuboid

A cuboid/rectangular prism is a three-dimensional shape with six rectangular faces. 

Cuboid

Its formulas are:

• Volume: V = lwh
• Surface area: A = 2lw + 2lh + 2wh
• Diagonal: d = √(l² + w² + h²), where l is the length, w is the width, and h is the height of the prism.

• Sphere

A sphere is a three-dimensional shape with a curved surface. 

Sphere

Its formulas are:

• Volume: V = (4/3)πr³
• Surface area: A = 4πr²
• Diameter: 2r, where r is the radius of the sphere.

• Cylinder

A cylinder is a three-dimensional shape with a circular base and a curved surface.

Cylinder

Its formulas are:

• Volume: V = πr²h
• Surface area: A = 2πr² + 2πrh
• Curved surface area: A_curved = 2πrh
• Base: The base of the cylinder is a circle with radius r.
• Height: The height of the cylinder is h.

• Cone

A cone is a three-dimensional shape with a circular base and a curved surface that tapers to a point.

Cone

Its formulas are:

• Volume: V = (1/3)πr²h
• Surface area: A = πr²+ πr√(r²+ h²)
• Curved surface area: A_curved = πr√(r²+ h²),
• Slant height: l = √(r²+ h²
• Base: The base of the cone is a circle with radius r.
• Height: The height of the cone is h.

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Co-ordinate Geometry Formulas

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The important formulas in coordinate geometry are as follows:

• Distance Formula: The distance between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

• Midpoint Formula: The midpoint between two points (x₁, y₁) and (x₂, y₂) in a coordinate plane is given by:

[(x₁ + x₂)/2, (y₁ + y₂)/2]

• Slope Formula: The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by:

m = (y₂ - y₁)/(x₂ - x₁)

• Equation of a Line: The equation of a line in slope-intercept form (y = mx + b) can be determined using the slope and the y-intercept (b). The slope (m) is given by the formula above, and the y-intercept (b) can be found using the point-slope formula:

y - y₁ = m(x - x₁)

• Parallel and Perpendicular Lines: Two lines are parallel if they have the same slope, and they are perpendicular if their slopes are negative reciprocals of each other. The equation of a line parallel to a given line can be found by keeping the same slope and changing the y-intercept, while the equation of a line perpendicular to a given line can be found by taking the negative reciprocal of the slope and changing the sign.

• Area of a Triangle: The area of a triangle formed by three points (x₁, y₁), (x₂, y₂) and (x₃, y₃) is given by:

A = 1/2 |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|

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Things to Remember

1. Geometry is a branch of mathematics that deals with shapes, sizes, and relative positions of objects.
2. It involves basic geometric shapes such as points, lines, angles, triangles, quadrilaterals, and circles, and important formulas and equations related to them such as perimeter, area, circumference, and diameter.
3. Concepts such as the Herons formula, Pythagorean theorem, congruence, and similarity of triangles are also important in class 9 mathematics.
4. Coordinate geometry is another branch of geometry that deals with the Cartesian plane and the distance and midpoint between two points.
5. Three-dimensional shapes such as cubes, cylinders, and spheres are also taught, along with their formulas related to volume and surface area.
6. Geometry has practical applications in fields such as engineering, architecture, and art.