2. Triangle Formulas
Comprehensive Triangle Formulas
| Formula Type |
Formula |
Description |
| Heron’s Formula |
A = √[s(s-a)(s-b)(s-c)] |
s = (a+b+c)/2 (semi-perimeter) |
| Pythagorean Theorem |
a² + b² = c² |
For right triangles; c = hypotenuse |
| Equilateral Triangle Area |
A = (√3/4) × a² |
a = side length |
| Isosceles Triangle Area |
A = (b/4) × √(4a² – b²) |
a = equal sides; b = base |
| Triangle with 2 sides & angle |
A = ½ × a × b × sin(C) |
C = included angle |
| Altitude |
h = (2A)/b |
h = altitude; b = base |
| Median Length |
m_a = ½√(2b² + 2c² – a²) |
Median to side a |
| Angle Sum |
∠A + ∠B + ∠C = 180° |
Sum of interior angles |
| Exterior Angle |
Exterior angle = sum of opposite interior angles |
Triangle exterior angle theorem |
Special Triangle Properties
| Triangle Type |
Side Ratios |
Angle Measures |
| 30-60-90 |
1 : √3 : 2 |
30°, 60°, 90° |
| 45-45-90 |
1 : 1 : √2 |
45°, 45°, 90° |
3. Quadrilateral Formulas
Detailed Quadrilateral Properties
| Quadrilateral |
Perimeter |
Area |
Special Properties |
| Square |
P = 4a |
A = a² = ½d² |
d = diagonal = a√2 |
| Rectangle |
P = 2(l + w) |
A = l × w |
Diagonal = √(l² + w²) |
| Parallelogram |
P = 2(a + b) |
A = b × h = ab sin(θ) |
θ = included angle |
| Rhombus |
P = 4a |
A = ½d₁d₂ = a²sin(θ) |
d₁, d₂ = diagonals |
| Trapezoid |
P = sum of all sides |
A = ½(a + b)h |
a, b = parallel sides |
| Kite |
P = 2(a + b) |
A = ½d₁d₂ |
d₁, d₂ = diagonals |
| General Quadrilateral |
P = a + b + c + d |
A = ½d₁d₂sin(θ) |
θ = angle between diagonals |
4. Circle Formulas
Complete Circle Geometry
| Component |
Formula |
Variables |
| Circumference |
C = 2πr = πd |
r = radius; d = diameter |
| Area |
A = πr² |
r = radius |
| Arc Length |
L = (θ/360°) × 2πr = rθ (radians) |
θ = central angle |
| Sector Area |
A = (θ/360°) × πr² = ½r²θ (radians) |
θ = central angle |
| Segment Area |
A = r²/2 (θ – sin θ) |
θ in radians |
| Chord Length |
c = 2r sin(θ/2) |
θ = central angle |
| Circle from Diameter |
(x-a)(x-b) + (y-c)(y-d) = 0 |
Endpoints: (a,c), (b,d) |
5. Coordinate Geometry Formulas (Class 10)
Essential Coordinate Geometry Formulas
| Formula |
Expression |
Application |
| Distance Formula |
d = √[(x₂-x₁)² + (y₂-y₁)²] |
Distance between two points |
| Section Formula (Internal) |
x = (m₁x₂ + m₂x₁)/(m₁+m₂), y = (m₁y₂ + m₂y₁)/(m₁+m₂) |
Point dividing line in ratio m₁:m₂ |
| Section Formula (External) |
x = (m₁x₂ – m₂x₁)/(m₁-m₂), y = (m₁y₂ – m₂y₁)/(m₁-m₂) |
External division in ratio m₁:m₂ |
| Midpoint Formula |
M = ((x₁+x₂)/2, (y₁+y₂)/2) |
Midpoint of line segment |
| Slope of Line |
m = (y₂-y₁)/(x₂-x₁) = tan(θ) |
θ = angle with x-axis |
| Area of Triangle |
A = ½ |
x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) |
| Collinearity Condition |
Area = 0 |
Three points are collinear |
| Centroid of Triangle |
G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) |
Center of mass |
Area Formula in Coordinate Geometry (Detailed)
Area of Triangle with Coordinates:
A = ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
Alternative Forms:
- Using determinant: A = ½ |det([[x₁, y₁, 1], [x₂, y₂, 1], [x₃, y₃, 1]])|
- Shoelace formula: A = ½ |(x₁y₂ + x₂y₃ + x₃y₁) – (y₁x₂ + y₂x₃ + y₃x₁)|
Area of Quadrilateral:
A = ½ |x₁(y₂-y₄) + x₂(y₃-y₁) + x₃(y₄-y₂) + x₄(y₁-y₃)|
6. Line Formulas in Coordinate Geometry
Equations of Lines
| Form |
Equation |
When to Use |
| Slope-Intercept |
y = mx + c |
m = slope; c = y-intercept |
| Point-Slope |
y – y₁ = m(x – x₁) |
Known: point and slope |
| Two-Point Form |
(y-y₁)/(y₂-y₁) = (x-x₁)/(x₂-x₁) |
Known: two points |
| Intercept Form |
x/a + y/b = 1 |
a = x-intercept; b = y-intercept |
| Normal Form |
x cos α + y sin α = p |
p = perpendicular distance from origin |
| General Form |
ax + by + c = 0 |
Standard form |
| Parametric Form |
x = x₁ + t cos θ, y = y₁ + t sin θ |
θ = angle; t = parameter |
Line Properties
| Property |
Formula |
| Parallel Lines |
m₁ = m₂ (equal slopes) |
| Perpendicular Lines |
m₁ × m₂ = -1 |
| Distance from Point to Line |
d = |
| Angle Between Lines |
tan θ = |
| Distance Between Parallel Lines |
d = |
7. 3D Geometry Formulas (Class 12)
Three-Dimensional Coordinate Geometry
| Formula |
Expression |
Description |
| Distance Formula (3D) |
d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²] |
Distance between two points |
| Section Formula (3D) |
x = (m₁x₂+m₂x₁)/(m₁+m₂), y = …, z = … |
Internal division |
| Midpoint (3D) |
M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) |
Midpoint formula |
| Direction Cosines |
l = a/r, m = b/r, n = c/r; l²+m²+n²=1 |
r = √(a²+b²+c²) |
| Direction Ratios |
a:b:c |
Proportional to direction cosines |
| Angle Between Vectors |
cos θ = (a₁a₂+b₁b₂+c₁c₂)/√(a₁²+b₁²+c₁²)√(a₂²+b₂²+c₂²) |
Dot product formula |
Equation of Line in 3D
| Form |
Equation |
| Cartesian Form |
(x-x₁)/a = (y-y₁)/b = (z-z₁)/c |
| Vector Form |
r⃗ = a⃗ + λb⃗ |
| Two-Point Form |
(x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁) |
Equation of Plane in 3D
| Form |
Equation |
| General Form |
ax + by + cz + d = 0 |
| Intercept Form |
x/a + y/b + z/c = 1 |
| Normal Form |
lx + my + nz = p |
| Vector Form |
r⃗ · n⃗ = d |
| Three-Point Form |
det([[x-x₁, y-y₁, z-z₁], [x₂-x₁, y₂-y₁, z₂-z₁], [x₃-x₁, y₃-y₁, z₃-z₁]]) = 0 |
3D Geometry Additional Formulas
| Concept |
Formula |
| Distance from Point to Plane |
d = |
| Angle Between Planes |
cos θ = |
| Distance Between Skew Lines |
d = |
| Shortest Distance |
For parallel lines: d = |
8. Surface Area and Volume Formulas (3D Shapes)
Complete 3D Geometry Formulas
| Shape |
Surface Area |
Volume |
Variables |
| Cube |
SA = 6a² |
V = a³ |
a = edge length |
| Cuboid |
SA = 2(lb + bh + hl) |
V = l × b × h |
l, b, h = dimensions |
| Sphere |
SA = 4πr² |
V = (4/3)πr³ |
r = radius |
| Hemisphere |
TSA = 3πr²; CSA = 2πr² |
V = (2/3)πr³ |
TSA = total; CSA = curved |
| Cylinder |
TSA = 2πr(r+h); CSA = 2πrh |
V = πr²h |
r = radius; h = height |
| Cone |
TSA = πr(r+l); CSA = πrl |
V = (1/3)πr²h |
l = slant height = √(r²+h²) |
| Frustum of Cone |
CSA = πl(r₁+r₂); TSA = π[l(r₁+r₂)+r₁²+r₂²] |
V = (1/3)πh(r₁²+r₂²+r₁r₂) |
r₁, r₂ = radii; h = height |
| Pyramid |
LSA = (1/2) × Perimeter × slant height |
V = (1/3) × Base Area × h |
h = vertical height |
| Prism |
LSA = Perimeter × h |
V = Base Area × h |
h = height |
| Torus |
SA = 4π²Rr |
V = 2π²Rr² |
R = major radius; r = minor radius |
9. Circle Theorems and Properties
Important Circle Theorems
| Theorem |
Formula/Property |
| Chord-Radius Theorem |
Perpendicular from center bisects chord |
| Tangent-Radius Theorem |
Tangent ⊥ radius at point of contact |
| Two Tangents |
Tangents from external point are equal in length |
| Inscribed Angle |
θ_inscribed = (1/2) × θ_central |
| Cyclic Quadrilateral |
Opposite angles sum to 180° |
| Power of Point |
PA × PB = PC × PD (intersecting chords) |
| Tangent-Secant |
(Tangent)² = External segment × Whole secant |
10. Trigonometric Formulas in Geometry
Right Triangle Trigonometry
| Ratio |
Formula |
Definition |
| sin θ |
Opposite/Hypotenuse |
O/H |
| cos θ |
Adjacent/Hypotenuse |
A/H |
| tan θ |
Opposite/Adjacent |
O/A |
| csc θ |
Hypotenuse/Opposite |
1/sin θ |
| sec θ |
Hypotenuse/Adjacent |
1/cos θ |
| cot θ |
Adjacent/Opposite |
1/tan θ |
Law of Sines and Cosines
| Law |
Formula |
| Law of Sines |
a/sin A = b/sin B = c/sin C = 2R |
| Law of Cosines |
c² = a² + b² – 2ab cos C |
| Law of Tangents |
(a-b)/(a+b) = tan[(A-B)/2]/tan[(A+B)/2] |
11. Vector Formulas in Geometry
Essential Vector Operations
| Operation |
Formula |
Description |
| Magnitude |
v⃗ |
| Unit Vector |
û = v⃗/ |
v⃗ |
| Dot Product |
a⃗ · b⃗ = |
a⃗ |
| Cross Product |
a⃗ × b⃗ |
| Scalar Triple Product |
[a⃗ b⃗ c⃗] = a⃗ · (b⃗ × c⃗) |
Volume of parallelepiped |
| Vector Triple Product |
a⃗ × (b⃗ × c⃗) = (a⃗·c⃗)b⃗ – (a⃗·b⃗)c⃗ |
BAC-CAB rule |
12. Transformation Geometry Formulas
Geometric Transformations
| Transformation |
Formula |
Effect |
| Translation |
(x’, y’) = (x+a, y+b) |
Shift by vector (a, b) |
| Reflection (x-axis) |
(x’, y’) = (x, -y) |
Mirror across x-axis |
| Reflection (y-axis) |
(x’, y’) = (-x, y) |
Mirror across y-axis |
| Rotation (origin) |
x’ = x cos θ – y sin θ; y’ = x sin θ + y cos θ |
Counterclockwise by θ |
| Scaling |
(x’, y’) = (kx, ky) |
Enlarge by factor k |
| Reflection (y=x) |
(x’, y’) = (y, x) |
Swap coordinates |
Frequently Asked Questions about Geometry Formula
Q. What is the area formula for a triangle in coordinate geometry?
The area of a triangle with vertices at (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated using:
A = ½ |x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)|
The absolute value ensures the area is always positive. This formula is derived from the determinant method and is especially useful when working with coordinate points rather than base and height.
Q. What is the section formula in coordinate geometry and when is it used?
The section formula helps find coordinates of a point that divides a line segment in a given ratio:
Internal Division (m₁:m₂):
- x = (m₁x₂ + m₂x₁)/(m₁ + m₂)
- y = (m₁y₂ + m₂y₁)/(m₁ + m₂)
External Division:
- x = (m₁x₂ – m₂x₁)/(m₁ – m₂)
- y = (m₁y₂ – m₂y₁)/(m₁ – m₂)
Use this when you need to find a point that divides a line segment between two known points in a specific ratio.
Q. What are the most important 3D geometry formulas for Class 12?
3D geometry formulas include:
- Distance formula: d = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
- Direction cosines: l² + m² + n² = 1
- Equation of line: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c
- Equation of plane: ax + by + cz + d = 0
- Angle between vectors: cos θ = (a₁a₂ + b₁b₂ + c₁c₂)/(|a⃗||b⃗|)
These are fundamental for solving problems involving three-dimensional space.
Q. How do I remember all geometry formulas easily?
Effective memorization strategies:
- Group by shape: Learn all circle formulas together, then triangles, etc.
- Understand derivations: Knowing why formulas work helps retention
- Practice regularly: Solve problems daily to reinforce memory
- Create flashcards: Write formula on one side, application on the other
- Use mnemonics: Create memory aids (e.g., “SOH-CAH-TOA” for trigonometry)
- Make a formula sheet: Handwrite all formulas as a study tool
- Visualize: Draw diagrams alongside formulas
Q. What’s the difference between 2D and 3D coordinate geometry?
2D Coordinate Geometry:
- Uses (x, y) coordinates on a plane
- Distance formula: √[(x₂-x₁)² + (y₂-y₁)²]
- Deals with lines, circles, and planar shapes
- Taught primarily in Class 10
3D Coordinate Geometry:
- Uses (x, y, z) coordinates in space
- Distance formula: √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²]
- Involves planes, lines in space, and spatial relationships
- Taught primarily in Class 12
The fundamental concepts are similar, but 3D adds complexity with the third dimension.
Q. What is Heron’s formula and when should I use it?
Heron’s Formula: A = √[s(s-a)(s-b)(s-c)], where s = (a+b+c)/2
Use Heron’s formula when:
- You know all three sides of a triangle
- You don’t know the height
- The triangle is scalene (all sides different)
Q. How do you find the equation of a line in coordinate geometry?
There are several forms depending on what information you have:
Given slope (m) and y-intercept (c):
Given point (x₁, y₁) and slope m:
Given two points (x₁, y₁) and (x₂, y₂):
- First find slope: m = (y₂ – y₁)/(x₂ – x₁)
- Then use point-slope form or: (y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁)
Choose the form based on your given information.
Q. What are the formulas for surface area and volume of common 3D shapes?
Most commonly used:
| Shape |
Volume |
Surface Area |
| Cube |
a³ |
6a² |
| Sphere |
(4/3)πr³ |
4πr² |
| Cylinder |
πr²h |
2πr(r + h) |
| Cone |
(1/3)πr²h |
πr(r + l) |
| Cuboid |
lbh |
2(lb + bh + hl) |
These formulas are essential for solving real-world problems involving capacity, packaging, and construction.
Q. What is the distance formula and how is it derived?
2D Distance Formula: d = √[(x₂ – x₁)² + (y₂ – y₁)²]
Derivation: Based on the Pythagorean theorem. The horizontal distance is (x₂ – x₁) and vertical distance is (y₂ – y₁). These form the legs of a right triangle, with the distance as the hypotenuse.
3D Distance Formula: d = √[(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²]
This extends the 2D formula by adding the third dimension (z-coordinate).
Q. How do you determine if three points are collinear using coordinate geometry?
Method 1 – Area Formula: Three points are collinear if the area of the triangle they form is zero:
- Calculate: ½ |x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)|
- If this equals 0, points are collinear
Method 2 – Slope Method: Points A, B, C are collinear if:
- Slope of AB = Slope of BC = Slope of AC
- Check: (y₂ – y₁)/(x₂ – x₁) = (y₃ – y₂)/(x₃ – x₂)
Method 3 – Distance Method:
- If AB + BC = AC (or any similar combination), points are collinear
Study Tips for Mastering Geometry Formulas
- Practice with diagrams: Always draw the shape when solving problems
- Understand units: Remember area is in square units, volume in cubic units
- Check your work: Verify answers make sense (areas and volumes can’t be negative)
- Start simple: Master 2D geometry before moving to 3D
- Connect concepts: Notice how formulas relate to each other
- Use technology: Graphing calculators and geometry software help visualization
- Solve previous years’ papers: Identify frequently tested formulas
- Create summary sheets: Organize formulas by topic for quick review
This comprehensive guide covers all essential geometry formulas from basic 2D shapes through advanced 3D coordinate geometry. Regular practice with these formulas, combined with understanding their derivations and applications, will build strong geometric problem-solving skills essential for academic success in mathematics.
Remember that geometry is visual always sketch diagrams, label all given information, and identify which formula applies before calculating. With consistent practice and proper understanding of these formulas, students can excel in geometry at all academic levels.
