Geometry Formulas: Area, Perimeter, Volume and Surface Area 2026-2027

Geometry formulas are equations used to calculate measurements of shapes, solids, angles, and coordinates. They help students find area, perimeter, volume, surface area, distance, and other values in geometry questions.

Most students lose marks not because they forgot the formula but because they picked the wrong one. A student who confuses curved surface area with total surface area, or uses diameter instead of radius, loses marks on a question they actually knew. This article organises all geometry formulas by shape and concept so students can find and apply them correctly.

Key Takeaways

Basic Geometry Formulas for Students

Basic geometry formulas for students start with 2D shapes. These formulas cover fields, tiles, walls, diagrams, graphs, and measurement problems that appear in every class from 6 to 10.

Geometry Formula for Square

A square has four equal sides and four right angles.

• Area = a²
• Perimeter = 4a
• Diagonal = a√2

Here, a means side length. If the side is 8 cm, area is 64 cm² and perimeter is 32 cm.

Geometry Formula for Rectangle

A rectangle has opposite sides equal and four right angles. It appears in questions based on rooms, boards, plots, and pages.

• Area = l × b
• Perimeter = 2(l + b)
• Diagonal = √(l² + b²)

Convert both values into the same unit before substituting.

Triangle Formulas

Triangle formulas depend on the information given. Some questions give base and height, others give all three sides.

• Area = 1/2 × b × h
• Perimeter = a + b + c
• Area of equilateral triangle = √3/4 × a²
• Height of equilateral triangle = √3/2 × a

Heron’s Formula for Triangle

Heron's formula applies when all three sides are known and height is not given.

• Semi-perimeter: s = (a + b + c)/2
• Area = √[s(s − a)(s − b)(s − c)]

Calculate semi-perimeter first, then substitute into the area formula.

Geometry Formulas for Area and Perimeter

Geometry formulas for area and perimeter solve most 2D shape questions. The question usually signals which one is needed. Words like covered, painted, or tiled point to area; words like fencing, border, or outline point to perimeter.

Area Formulas in Geometry

Area formulas in geometry always produce square units: cm², m², km².

• Square: a²
• Rectangle: l × b
• Triangle: 1/2 × b × h
• Parallelogram: b × h
• Rhombus: 1/2 × d₁ × d₂
• Trapezium: 1/2 × (a + b) × h
• Circle: πr²

Perimeter Formulas in Geometry

Perimeter formulas always produce linear units: cm, m, km.

• Square: 4a
• Rectangle: 2(l + b)
• Triangle: a + b + c
• Parallelogram: 2(a + b)
• Rhombus: 4a
• Circle (circumference): 2πr

Circle Formulas in Geometry

Circle formulas appear in questions on tracks, sectors, shaded regions, and arcs. Always check whether the question gives radius or diameter before substituting.

Circle Formulas for Radius, Diameter and Circumference

A circle is a closed curve where every point stays at the same distance from the centre.

• Diameter = 2r
• Radius = d/2
• Circumference = 2πr
• Area = πr²

If diameter is given, divide by 2 before using area or circumference formulas.

Circle Formulas for Arc and Sector

An arc is a part of the circumference. A sector is the region between two radii and an arc.

• Arc length = θ/360 × 2πr
• Area of sector = θ/360 × πr²

Here, θ is the central angle in degrees.

Mensuration Formulas for 3D Shapes

Mensuration formulas help students calculate volume and surface area of solids. These formulas are useful for cubes, cuboids, cylinders, cones, spheres, and hemispheres.

Geometry Formulas for 3D Shapes

Geometry formulas for 3D shapes help students solve questions on containers, boxes, pipes, balls, tanks, and solid objects. TSA means total surface area. CSA means curved surface area. l in cone formulas means slant height.

• Volume of cube = a³ | TSA = 6a²
• Volume of cuboid = l × b × h | TSA = 2(lb + bh + lh)
• Volume of cylinder = πr²h | CSA = 2πrh | TSA = 2πr(r + h)
• Volume of cone = 1/3πr²h | CSA = πrl | TSA = πr(l + r)
• Volume of sphere = 4/3πr³ | SA = 4πr²
• Volume of hemisphere = 2/3πr³ | TSA = 3πr²

Volume Formulas in Geometry

Volume formulas measure the space inside a solid. Use cubic units: cm³, m³, or litres.

• Cuboid: length × breadth × height
• Cylinder: πr²h
• Cone: 1/3πr²h
• Sphere: 4/3πr³

Convert all dimensions into the same unit before substituting.

Surface Area Formulas

Surface area formulas help in questions about painting, wrapping, polishing, coating, or covering an object. The question decides whether students need CSA, TSA, or lateral surface area.

Curved surface area covers only the curved part of a solid. Total surface area covers every exposed surface. For open containers, students should exclude the missing surface.

For example, a closed cylinder needs total surface area. An open cylindrical container usually needs curved surface area plus base area.

Triangle Formulas for Angles and Sides

Triangle formulas also include angle rules and side relationships. These formulas help in both numerical and theorem-based questions.

Angle Formulas for Triangle

Every triangle has three angles. Their sum always equals 180°.

∠A + ∠B + ∠C = 180°

Exterior angle = Sum of two opposite interior angles

In an equilateral triangle, each angle is 60°. In a right-angled triangle, one angle is 90°.

Pythagoras Theorem Formula

Pythagoras theorem applies only to right-angled triangles. It connects the hypotenuse with the other two sides.

a² + b² = c²

Here, c is the hypotenuse. It is the side opposite the right angle.

Use this formula when two sides of a right triangle are given and the third side must be found.

Polygon Geometry Formulas

Polygon geometry formulas help students solve angle-based questions for figures with many sides. These formulas work for triangles, quadrilaterals, pentagons, hexagons, and regular polygons.

Interior Angle Sum Formula

A polygon with n sides has this interior angle sum:

Sum of interior angles = (n - 2) × 180°

For a quadrilateral, n = 4. So, the sum of interior angles is 360°.

Each Interior Angle of a Regular Polygon

A regular polygon has equal sides and equal angles. Use this formula when the question says regular polygon.

Each interior angle = [(n - 2) × 180°]/n

For a regular hexagon, n = 6. Each interior angle is 120°.

Exterior Angle Formula

The sum of exterior angles of any polygon is always 360°.

Each exterior angle of a regular polygon = 360°/n

This formula helps when students know the number of sides or one exterior angle.

Coordinate Geometry Formulas

Coordinate geometry formulas connect points, lines, and shapes with the number plane. They are useful when questions give ordered pairs such as (x₁, y₁) and (x₂, y₂).

Distance Formula

The distance formula gives the length between two points on the coordinate plane.

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This formula comes from Pythagoras theorem. It helps find the distance between two points without drawing an accurate diagram.

Midpoint Formula

The midpoint formula gives the centre point of a line segment.

Midpoint = [(x₁ + x₂)/2, (y₁ + y₂)/2]

Use it when the question asks for the point exactly halfway between two given points.

Section Formula

The section formula gives the point that divides a line segment in a given ratio.

Point = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)]

Use this formula when a point divides a line segment internally in the ratio m:n.

Area of Triangle Formula in Coordinate Geometry

The area of a triangle with three coordinate points is:

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

If the area becomes zero, the three points are collinear. This means they lie on the same straight line.

How to Choose the Correct Geometry Formula

Read the question slowly. The shape, given values, and final requirement point to the formula.

• Boundary → perimeter
• Covered region → area
• Space inside a solid → volume
• Painting or wrapping → surface area
• Circle question → check radius or diameter first
• 3D shape → check open or closed before choosing TSA or CSA
• Coordinate question → label points before substituting

Solved Examples on Geometry Formulas

Solved examples help students apply formulas correctly. They also show how units change with area, perimeter, and volume.

Example 1: Area of Rectangle

Question: Find the area of a rectangle with length 12 cm and breadth 5 cm.

Formula: Area = l × b

Solution: Area = 12 × 5 = 60 cm²

Answer: The area of the rectangle is 60 cm².

Example 2: Circumference of Circle

Question: Find the circumference of a circle with radius 7 cm.

Formula: Circumference = 2πr

Solution: Circumference = 2 × 22/7 × 7 = 44 cm

Answer: The circumference is 44 cm.

Example 3: Volume of Cuboid

Question: Find the volume of a cuboid with length 8 cm, breadth 4 cm, and height 3 cm.

Formula: Volume = l × b × h

Solution: Volume = 8 × 4 × 3 = 96 cm³

Answer: The volume of the cuboid is 96 cm³.

Geometry Formulas List for Quick Revision

A geometry formulas list helps students revise faster before solving questions. Use this section when you need the correct formula quickly.