Area is the total amount of flat space inside the boundaries of a two-dimensional shape.
If you imagine painting a flat wall or covering a bedroom floor with a carpet, the total amount of surface space you cover is the area.
The official SI unit (International System of Units) for measuring area is the square meter (m2).
Because area measures a flat surface instead of a straight line path, it is always expressed in square units.
Common units include square millimeters (mm2), square centimeters (cm2), square meters (m2), and square kilometers (km2).
You must never use linear units like cm or m for area, because regular linear units are strictly reserved for measuring boundary lengths like perimeter, not flat surface space.
Area is a tool used to measure spaces in daily life.
Daily Examples:
To find the area of a rectangle shape, multiply its straight length side by its width side.
Problem:
A rectangular floor has a length of 6 meters and a width of 4 meters. Find the total flat area inside the floor.
Explanation:
Step 1: Write down the formula for the area of a rectangle.
Area = Length × Width
Step 2: Substitute the side lengths into the formula.
Area = 6m × 4m
Step 3: Multiply the numbers together.
6 × 4 = 24
Final Answer = 24m2
This section covers flat shapes made with straight lines. To find the area of these shapes, you use math formulas that calculate the flat space inside their outside walls. Below are the steps, diagrams, and examples for each straight shape.
A triangle is a three-sided shape. To find its area, multiply the flat horizontal base line by the perpendicular height, then divide the answer by 2.
Formula :
For a triangle with base b and perpendicular height h ;
Area = 1 2 × b × h
Example: Find the flat area inside the triangle shown above with a base of 8 cm and a perpendicular height of 5 cm.
Formula:
Area = 1 2 × b × h
b = 8 cm
h = 5 cm
Calculation:
Area = 1 2 × 8 × 5
Area = 4 × 5 = 20 cm2
A square is a four-sided shape with four identical, equal sides. You calculate its flat area by multiplying one side length by itself.
Formula :
For a square with side S ;
Area = S × S
Example: Find the flat area inside a square floor where one side length (S) is 6 meters.
Formula:
Area = S × S
S = 6 m
Area = 6 × 6 = 36 m2
A rectangle has two equal opposite lengths and two equal opposite widths. You find its flat surface space by multiplying the length side by the width side.
Formula :
For a rectangle with length L and width W ;
Area = L × W
Example: Find the flat area inside a rectangle window with length (L) 8 cm and width (W) 3 cm.
Formula:
Area = L × W
L = 8 cm
W = 3 cm
Calculation:
Area = 8 × 3
Area = 24 cm2
A parallelogram is a four-sided shape in which both pairs of opposite sides are parallel and equal in length. Its area is calculated by multiplying its flat horizontal base line by its straight perpendicular height line. Note that the slanted side is never used for area.
Formula :
For a parallelogram with base b and perpendicular height h ;
Area = b × h
Example: Find the flat area inside a parallelogram tile with a base of 9 mm and a perpendicular height of 4 mm.
Formula:
Area = b × h
b = 9 mm
h = 4 mm
Calculation:
Area = 9 × 4
Area = 36 mm2
A trapezium is a quadrilateral with at least one pair of parallel sides. To find its flat area, add the lengths of the two parallel sides (a and c) together, multiply by the perpendicular height (h), and then divide the entire answer by 2.
Formula :
For a trapezium with parallel sides a, c and perpendicular height h ;
Area = 1 2 (a + c) h
Example: Find the flat area inside the trapezium shown above with parallel sides of 4 cm and 10 cm, and a perpendicular height of 4 cm.
Formula:
Area = 1 2 × (a + c) × h
a = 4 cm
c = 10 cm
h = 4 cm
Calculation:
Area = 1 2 × (4 + 10) × 4
Area = 1 2 × 14 × 4
Area = 7 × 4 = 28 cm2
A rhombus is a four-sided shape with all four sides equal in length. To find its flat area, multiply the diagonal lines that cut through the center from corner to corner, then divide the answer by 2.
Formula :
For a rhombus with diagonals d1 and d2 ;
Area = 1 2 d1 d2
Example: Find the flat area inside a rhombus sign with a vertical diagonal (d1) of 12 cm and a horizontal diagonal (d2) of 6 cm.
Formula:
Area = 1 2 × d1 × d2
d1 = 12 cm
d2 = 6 cm
Calculation:
Area = 1 2 × 12 × 6
Area = 6 × 6 = 36 cm2
Multi-sided flat shapes built purely from straight lines joined corner-to-corner are called polygons. These boundaries follow strict addition or multiplication properties depending on their unique side patterns.
A kite is a four-sided shape with two pairs of equal-length adjacent sides. To calculate its flat area space, multiply the vertical and horizontal diagonal lines cutting through its center from corner to corner, then divide your answer by 2.
Formula:
For a kite with diagonals d1 and d2 ;
Area = 1 2 d1 d2
Example: Find the flat area inside a kite frame where the vertical diagonal (d1) is 10 cm and the horizontal diagonal (d2) is 6 cm.
d1 = 10 cm
d2 = 6 cm
Calculation:
Area = 1 2 × 10 × 6
Area = 5 × 6 = 30 cm2
A polygon is regular if every single one of its outer side lines is exactly the same length, and all of its interior corners match perfectly. To find its flat area space, you find the area of one inner triangular segment from the center point, then multiply by the total number of matching sides.
Formula:
For a regular shape with n number of sides, side base length s, and center vertical height distance (apothem) a ;
Area = 1 2 × n × s × a
Example: Find the flat area inside a regular 6-sided hexagon where each side (s) is 8 meters and the center height line (a) is 7 meters.
n = 6 sides
s = 8 m
a = 7 m
Calculation:
Area = 1 2 × 6 × 8 × 7
Area = 3 × 8 × 7 = 24 × 7 = 168 m2
An irregular polygon is a polygon whose sides and/or interior angles are not all equal. Because there is no consistency, no shortcut formula layout exists. You must split the irregular polygon layout into standard simpler shapes (like rectangles and triangles), find each flat area space independently, and add them together.
Formula:
Area = Area of Shape 1 + Area of Shape 2 + Area of Shape 3
Example: An irregular compound L-shaped flat lot is split into a square of side 5m and a rectangle measuring 6m by 4m. Find its total inside area.
Area of Square = 5m × 5m = 25m2
Area of Rectangle = 6m × 4m = 24m2
Calculation:
Total Area = 25 + 24
Total Area = 49 m2
This section covers flat shapes made with curved lines. To find the flat area space inside a curved shape, you measure the surface area enclosed within its round outer boundary using the radius or diameter. Below are the steps, diagrams, and examples for each curved shape.
The area of a circle is the amount of flat space enclosed by its circumference. It measures the flat surface enclosed by the circumference. You find this value by multiplying the mathematical constant Pi (π) by the radius squared (radius multiplied by itself).
Formula:
Area = πr2
*(Note: Remember that r2 means radius × radius, not radius × 2)*
Example: Find the flat inside area of a round dinner plate with a radius (r) of 7 cm.
Formula:
Area = πr2
Calculation:
Area = 22 7 × 7 × 7
Area = 22 × 1 × 7
Area = 22 × 7
Area = 154 cm2
A semicircle is exactly half of a perfect circle. To make the work easy, you can find its flat area space by multiplying a half fraction directly by Pi and the radius squared.
Formula:
Area = 1 2 πr2
Example: Find the flat inside area of a solid semicircle window with a radius (r) of 7 meters.
Formula:
Area = 1 2 πr2
r = 7 m
Calculation:
Area = 1 2 × 22 7 × 7 × 7
Area = 11 × 1 × 7
Area = 11 × 7
Area = 77 m2
A quadrant is a quarter section of a circle (one-fourth of a full circle). To find its flat area space smoothly, multiply a quarter fraction directly by Pi and the radius squared.
Formula:
Area = 1 4 πr2
Example: Find the flat inside area of a quadrant plate layout that has a radius (r) of 14 cm.
Step 1: Write down the quadrant area formula.
Area = 1 4 πr2
r = 14 cm
Calculation:
Area = 1 4 × 22 7 × 14 × 14
Area = 1 4 × 616
Area = 616 ÷ 4 = 154 cm2
Composite curved shapes combine rectangular straight segments and curved circular segments together. To find their total flat area, split the shape layout into distinct parts, calculate each surface space, and add them together.
Formula:
Area = Area of Rectangle + Area of Combined Circular Parts
Example: A stadium floor layout features a central rectangle with a length of 10 meters and a width of 14 meters, capped on both matching ends by two semicircle loops with a diameter of 14 meters. Find the total inside area space.
Step 1: Calculate the area of the central rectangle.
Area of the rectangle = Length × Width
Area of the rectangle = 10m × 14m = 140 m2
Step 2: Combine the two curved semicircle caps to form one full circle area.
The two semicircular arcs together form one complete circle. With a diameter of 14 m, the Radius (r) = 7 m.
Circle Area = πr2
Circle Area = 22 7 × 7 × 7 = 154 m2
Step 3: Add both separate area components together.
Total Area = Rectangle Area + Circle Area
Total Area = 140 m2 + 154 m2 = 294 m2
Example 1 (Rectangle Office Floor)
An office floor is shaped like a perfect rectangle. It has a length of 9.5 meters and a width of 4 meters. Calculate the total flat surface area inside the office floor.
Solution:
Step 1: Identify the given properties and note down the rectangle area formula.
Length (L) = 9.5 m
Width (W) = 4 m
Formula: Area = L × W
Step 2: Multiply the length by the width to find the flat inside space.
Area = 9.5 × 4
Area = 38 m2
Final Answer = 38 m2
Example 2 (Triangle Metal Plate)
A flat triangular metal plate has a horizontal base measuring 12 centimeters and a perpendicular height measuring 6 centimeters. Calculate the total surface area inside the plate.
Solution:
Step 1: Write down the fraction-style triangle area formula.
Area = 1 2 bh
Base (b) = 12 cm
Height (h) = 6 cm
Step 2: Substitute the values into the formula and complete the multiplication layout.
Area = 1 2 × 12 × 6
Area = 6 × 6
Area = 36 cm2
Final Answer = 36 cm2
Example 3 (Semicircular Garden Plot)
A garden flower bed is laid out in the shape of a perfect semicircle with an inner radius of 14 meters. Calculate the flat surface area space inside this garden bed.
Solution:
Step 1: Note down the semicircle area formula utilizing the fractional multiplier notation.
Area = 1 2 πr2
Radius (r) = 14 m, Pi (π) = 22/7
Step 2: Set up the simplified parameter expansion calculation step.
Area = 1 2 × 22 7 × 14 × 14
Area = 11 × 2 × 14
Area = 22 × 14
Area = 308 m2
Final Answer = 308 m2
Example 4 (Trapezium Metal Plate)
A custom-cut flat metal sheet is shaped like a trapezium. It features two parallel top and bottom bases measuring 6 centimeters and 12 centimeters respectively, with a perpendicular height line running through its center measuring 5 centimeters. Calculate the total flat surface area inside the sheet.
Solution:
Step 1: Write out the fraction-style trapezium area formula layout.
Area = 1 2 (a + c) h
Top Base (a) = 6 cm
Bottom Base (c) = 12 cm
perpendicular Height (h) = 5 cm
Step 2: Substitute the values into the brackets and simplify the terms.
Area = 1 2 × (6 + 12) × 5
Area = 1 2 × 18 × 5
Area = 9 × 5 = 45 cm2
Final Answer = 45 cm2
Example 5 (Regular Hexagon Floor Emblem)
A decorative regular 6-sided hexagon is set into a floor layout. Each flat outer side measures exactly 10 centimeters and its perpendicular center line distance (apothem) measures 8.66 centimeters. Calculate the total flat surface space covered by the emblem.
Solution:
Step 1: Set up the regular polygon area calculation formula using standard parameters.
Area = 1 2 × n × s × a
Total Sides (n) = 6
Side Base (s) = 10 cm
Apothem Height (a) = 8.66 cm
Step 2: Substitute the values into the equation framework to compute the total area region.
Area = 1 2 × 6 × 10 × 8.66
Area = 3 × 10 × 8.66
Area = 30 × 8.66
Area = 259.8 cm2
Final Answer = 259.8 cm2
Attempt the following questions to test your understanding of area concepts:
(1.) A square room floor has a single wall measurement of exactly 7 meters. Calculate the total flat surface area inside the room.
(2.) A flat triangular highway sign has a horizontal base line of 10 centimeters and a perpendicular height of 12 centimeters. Find the total area inside the sign.
(3.) A large circular dining table layout has a radius measuring exactly 14 meters. Calculate the flat inside area space covered by the table.
(4.) A flat sheet of metal is shaped like a trapezium. It has two parallel bases measuring 5 centimeters and 9 centimeters, with a perpendicular height of 6 centimeters. Determine its total surface area.
(5.) A regular 6-sided hexagon tile has a single outer side length of 12 centimeters and a center vertical height distance (apothem) of 10.4 centimeters. Calculate the total flat surface area inside the tile.
Take Note:
We keep our tools and mathematics resources completely free for everyone. If our platform helps you excel, consider supporting our work.