GEOMETRY

Area and Volume

Area measures the amount of flat space inside a two-dimensional shape.

Volume measures the amount of space inside a three-dimensional object.

Area

Area is used for 2D shapes and involves two dimensions, such as length and width.

Units

- Measured in square units.

- Examples: cm², m², km²

Common Formulas

Rectangle

A = length × width

Triangle

A = ½ × base × height

Circle

A = πr²

- r = radius

- π ≈ 3.14

Example use: finding how much carpet is needed to cover a floor.

Volume

Volume is used for 3D objects and involves three dimensions, such as length, width, and height.

Units

- Measured in cubic units.

- Examples: cm³, m³, km³

Common Formulas

Cube

V = side³

Rectangular Prism

V = length × width × height

Cylinder

V = πr²h

Sphere

V = 4/3πr³

- r = radius

- h = height

- π ≈ 3.14

Example use: finding how much water a tank can hold.

Quick Review

Area → space inside a 2D shape → square units (²)

Volume → space inside a 3D object → cubic units (³)

Rectangle → l × w

Triangle → ½bh

Circle → πr²

Cube → s³

Rectangular prism → lwh

Cylinder → πr²h

Sphere → 4/3πr³

Angles

An angle is formed when two rays meet at a common point called the vertex.

Parts of an Angle

Vertex

- The common endpoint where the two rays meet.

- Forms the corner of the angle.

Arms or Sides

- The two rays that extend from the vertex.

Degrees (°)

- The standard unit used to measure the size of an angle.

Types of Angles

Acute Angle

- Measures less than 90°.

- Example: 45°

Right Angle

- Measures exactly 90°.

- Forms an L shape.

Obtuse Angle

- Measures more than 90° but less than 180°.

- Example: 120°

Straight Angle

- Measures exactly 180°.

- Forms a straight line.

Reflex Angle

- Measures more than 180° but less than 360°.

- Example: 270°

Full Rotation

- Measures exactly 360°.

- Forms a complete circle.

Quick Review

Acute → < 90°

Right → = 90°

Obtuse → 90° < angle < 180°

Straight → = 180°

Reflex → 180° < angle < 360°

Full rotation → = 360°

Perimeter

The perimeter is the total distance or length around the outside of a closed two-dimensional shape.

Basics

- Measures the outer boundary of a shape.

- Uses linear units.

- Examples: cm, m, km, inches, miles.

- To find the perimeter, add the lengths of all sides.

Common Formulas

Square

P = 4s

- s = side length

Rectangle

P = 2(l + w)

- l = length

- w = width

Triangle

P = a + b + c

- Add the lengths of all three sides.

Circle

The perimeter of a circle is called its circumference.

C = 2πr

- r = radius

- π ≈ 3.14

Quick Review

Perimeter → distance around a shape

Units → linear units (cm, m, km, etc.)

Square → 4s

Rectangle → 2(l + w)

Triangle → a + b + c

Circle → 2πr (circumference)