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## Area of Plane Shapes

### Example: What is the area of this rectangle?

### Example: What is the area of this circle?

### Example: What is the area of this triangle?

A harder example:

### Example: Sam cuts grass at $0,10 per square meter

### How much does Sam earn cutting this area:

### Perimeter, Area, and Volume

1. The **perimeter **of a polygon (or any other closed curve, such as a circle) is the distance around the outside.

2. The **area **of a simple, closed, planar curve is the amount of space inside.

3. The **volume **of a solid 3D shape is the amount of space displaced by it.

Some formulas for common 2 -dimensional plane figures and 3 -dimensional solids are given below. The answers have one, two, or three dimensions; **perimeter **is measured in **linear units **, **area **is measured in **square units **, and **volume **is measured in **cubic units**.

### FORMULAS FOR PERIMETER, AREA, SURFACE, VOLUME

### Area Of Shapes

In Geometry, a shape is defined as the figure closed by the boundary. The boundary is created by the combination of lines, points and curves. Basically, there are two different types of geometric shapes such as:

- Two – Dimensional Shapes
- Three – Dimensional Shapes

Each and every shape in the Geometry can be measured using different measures such as area, volume, surface area, perimeter and so on.

### What is Area?

An area is a quantity that expresses the extent of a two-dimensional figure or shape or planar lamina in the plane. Lamina shapes include 2D figures that can be drawn on a plane, e.g., circle, square, triangle, rectangle, trapezium, rhombus and parallelogram.** Area of shapes** such as circle, triangle, square, rectangle, parallelogram, etc. are the region occupied by them in space.

**Polygon shape: **A polygon is a two-dimensional shape that is formed by straight lines. The examples of polygons are triangles, hexagons and pentagons. The names of shapes describe how many sides exist in the shape. For instance, a triangle consists of three sides and a rectangle has four sides. Hence, any shape that can be formed using three straight lines is known as a triangle and any shape that can be drawn by linking four lines is known as a **quadrilateral**. The area is the region inside the boundary/perimeter of the shapes which is to be considered.

### What are 2D shapes?

The two-dimensional shapes (2D shapes) are also known as flat shapes, are the shapes having two dimensions only. It has length and breadth. It does not have thickness. The two different measures used for measuring the flat shapes are area and the perimeter. Two-dimensional shapes are the shapes that can be drawn on the piece of paper. Some of the examples of 2D shapes are square, rectangle, circle, triangle and so on.

### Area of 2D Shapes Formula

In general, the area of shapes can be defined as the amount of paint required to cover the surface with a single coat. Following are the ways to calculate area based on the number of sides that exist in the shape, as illustrated below in the fig.

Let us write the formulas for all the different types of shapes in a tabular form.

Shape |
Area |
Terms |

Circle | π × r^{2} |
r = radius of the circle |

Triangle | ½ × b × h | b = base
h = height |

Square | a^{2} |
a = length of side |

Rectangle | l × w | l = length
w = width |

Parallelogram | b × h | b=base
h=vertical height |

Trapezium | ½(a+b) × h | a and b are the length of parallel sides
h = height |

Ellipse | πab | a = ½ minor axis
b = ½ major axis |

### What are 3D shapes?

The three-dimensional shapes (3D shapes), known as solid shapes, are the shapes that have three dimensions such as length, breadth and thickness. The two distinct measures used to define the three-dimensional shapes are volume and surface area. Generally, the three-dimensional shapes are obtained from the rotation of two-dimensional shapes. Thus, the surface area of any 2D shapes should be a 2D shape. If you want to calculate the surface area of solid shapes, we can easily calculate from the area of 2D shapes.

### Area of 3D Shapes Formula

According to the International System of Units (SI), the standard unit of area is the square meter (written as m^{2}) and is the area of a square whose sides are one meter long. For example, a particular shape with an area of three square meters would have the same area as three such squares. The surface area of a solid object is a measure of the total area that the surface of the object occupies.

For 3D/ solid shapes like cube, cuboid, sphere, cylinder and cone, the area is updated to the concept of the surface area of the shapes. The formulas for three-dimension shapes are given in the table here:

Shape |
Surface area |
Terms |

Cube | 6a^{2} |
a = length of the edge |

Rectangular prism | 2(wl+hl+hw) | l = length
w = width h = height |

Cylinder | 2πr(r + h) | r = radius of circular base
h = height of the cylinder |

Cone | πr(r + l) | r = radius of circular base
l = slant height |

Sphere | 4πr^{2} |
r = radius of sphere |

Hemisphere | 3πr^{2} |
r = radius of hemisphere |

In addition to the area of the planar shapes, an additional variable i.e the height or the radius are taken into account for computing the surface area of the shapes.

Consider a circle of radius r and make endless concentric circles. Now from the centre to the boundary make a line segment equal to the radius and cut the figure along with that segment. It’ll be formed a triangle with base equal to the circumference of the circle and height is equal to the radius of the outer circle, i.e., r. The area can thus be calculated as ½ * base * height i.e

½ * 2πr*r

### What Are Area Formulas?

Different **Area Formulas** are used to calculate the area of different geometric figures. A few of the important geometric figures are square, rectangle, circle, triangle, trapezoid, ellipse.

### Area Formula For Triangle:

The area of a triangle can be calculated by finding the half of the product of the length of its base and height. The area of a triangle formula is given as,

Area of a Triangle = 1/2 × Base × Height

### Area Formula For Square:

The area of a square is calculated by finding the square of the length of square’s each side. The area of a square formula is given as,

Area of Square = Side × Side = Side^{2}

### Area Formula For Circle:

The area of the circle can be calculated b finding the product of pi with the square of the radius. The area of a circle formula is given as,

Area of a Circle = πr^{2}

### Area Formula For Rectangle:

The area of a rectangle is calculated by multiplying the length and breadth of a rectangle. The area of a rectangle formula is given as,

Area of Rectangle = Length × Breadth

### Area Formula For Parallelogram:

The area of a parallelogram can be calculated by multiplying the base and height of a parallelogram. The area of a parallelogram formula is given as,

Area of a Parallelogram = Base × Height

### Area Formula For a Rhombus:

The area of a rhombus can be calculated by finding half of the product of lengths of its diagonals. The area of a quadrilateral formula is given as,

Area of a Rhombus = 1/2 × Diagonal_{1} × Diagonal_{2}

### Area Formula For a Trapezoid:

The area of a trapezoid is half of the product of sum of parallel sides and height. The area of a trapezoid formula is given as,

Area of a Trapezoid = 1/2 × Sum of Parallel Sides (a + b) × Height(h)

### Area Formula For Ellipse:

The area of an ellipse is the product of pi with a major axis and minor axis. The area of an ellipse formula is given as,

Area of Ellipse = π × Major axis(a) × Minor Axis(b)

### Area of Shapes Examples

**Example 1:**

Find the area of the circular path whose radius is 7m.

**Solution:**

Given, radius of circular path, r = 7m

By the formula of area of circle, we know;

A = π r^{2}

A = 22/7 x 7 x 7

A = 154 sq.m.

**Example 2:**

The side-length of a square plot is 5m. Find the area of a square plot.

**Solution:**

Given, side length, a = 5m

By the formula of area of a square, we know;

Area = a^{2}

A = 5 x 5

A = 25 sq.m.

**Example 3:**

Find the area of the cone, whose radius is 4cm and height is 3cm.

**Solution:**

Given, radius of cone = 4cm

and height of cone = 3cm.

As per the formula of area of cone, we know;

Slant height =l = √(4^{2} + 3^{2}) = √25 = 5 cm

Area = πr(r + l)

= (22/7) × 4(4 + 5)

=(22/7) 36

= 113. 14 cm^{2}

### Examples Using Geometric Area Formula

**Example 1:** What is the area of a rectangular park whose length and breadth are 60m and 90m respectively?

**Solution: **

To find: The area of a rectangular park.

Given:

Length of the park = 60m

Breadth of the park = 90m

Using area formula,

Area of Rectangle = (L × B)

= (60 × 90) m^{2 }

= 5400 m^{2}

**Answer: **The area of the rectangular park is 5400 m^{2}.

**Example 2**: What is the area of a circular park whose radius is 400m?

**Solution: **

To find: The area of a circular park.

Given:

Radius of the circular park = 400m

Using area formula,

Area of a Circle = πr^{2}

Area of a Circle = π 400^{2}

= 160000π m^{2}

**Answer: **The area of the circular park is 160000π m^{2}.

**Example 3: **What is the area of a square with each side measuring 5 units?

**Solution: **

We have, using area formula for square,

Area = 5 × 5 = 25 square units

**Answer:** The area of square is 25 square units.

**FAQs on Area Formulas**

#### What Are Area Formulas?

Area formulas are used to find the area or region covered by any 2-D shape. Area formula for few shapes is given as,

- Area of square = (side)
^{2} - Area of rectangle = length × breadth
- Area of triangle = (1/2) × base × height

#### What Is the Area Formula For Square?

The area formula for square is used to find the area covered by a square in 2-D plane. The area formula for square is given as, Area of square = (side)^{2}.

#### What Is the Area Formula For Rectangle?

The area formula for the rectangle is used to find the area covered by a rectangle in 2-D plane. The area formula for the rectangle is given as, Area of rectangle = length × breadth.

#### What Is the Application of Area Formula?

Area formula finds application in calculating the total region covered by a shape in 2-D plane. Different area formulas can be used to calculate area of different depending upon the type and parameters given.

**MATHS Related Links**

Area of Regular Polygon Formula

✅ Area Of Isosceles Triangle ⭐️⭐️⭐️⭐️⭐️

✅ Area Of An Octagon Formula ⭐️⭐️⭐️⭐️⭐

✅ Area Of An Octagon Formula ⭐️⭐️⭐️⭐️⭐

✅ Area of a Pentagon Formula ⭐️⭐️⭐️⭐️⭐

Surface Area of a Cube Formula

Surface Area of a Prism Formula

Surface Area of a Rectangular Prism Formula

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