Area of Geometric Figures – Formulas and Diagrams

The area of geometric figures represents the region covered by the figure in space. The area is one of the most important measurements of geometric figures. Area is a two-dimensional measure, so it has square units, such as m², cm², and so on. We can calculate the area of both two-dimensional figures and three-dimensional figures. The formula for the area depends on the shape of the figure and its dimensions.

Here, we will learn about the formulas for the areas of the most important geometric figures. Then, we will apply these formulas to solve some problems.

GEOMETRY
formula for the area of a pentagon

Relevant for

Learning about the area of geometric figures.

See formulas

GEOMETRY
formula for the area of a pentagon

Relevant for

Learning about the area of geometric figures.

See formulas

Definition of area

The area of a geometric figure is defined as the region occupied by a figure in space. Area is one of the most important measurements of geometric figures. Since area is a two-dimensional measure, we use square units to measure it. For example, we can use m², cm², and so on.

The area can be calculated from a 2D figure, as well as from a 3D figure. In the case of the 3D figure, the area is called the surface area and it is the area of all the faces of the figure. For two figures to have the same area, their shape and dimensions must be the same.


Formulas for the area of 2D figures

The area formulas depend on the shape of the 2D figure. The most important 2D figures are the square, triangle, rectangle, circle, parallelogram, trapezium, and ellipse.

Area of a rectangle

A rectangle is a figure that has four 90° interior angles. The opposite sides of a rectangle are the same length.

rectangle with dimensions

Area of rectangle = a×b

where, a represents the width of the rectangle and b represents its length.

Area of a square

A square is defined as a special type of rectangle that has all its sides the same length.

diagram of a square with dimensions

Area of a square = l²

where l represents the length of one of the sides of the square.

Area of a triangle

A triangle is defined as a three-sided polygon. There are three main types of triangles: equilateral, isosceles, and scalene. However, the area formula applies to any type of triangle.

diagram of a triangle with sides and height

Area of a triangle = ½ha

where, h represents the height and a represents the base of the triangle.

Area of a circle

The circle is defined as the set of points that are located at the same distance from a central point. That distance is called the radius and is used to calculate the area of the circle.

diagram of a circle with radius

Area of a circle = πr²

where r represents the radius and π is a numerical constant with an approximate value of 3.1415…

Area of a parallelogram

The parallelogram is a figure in which its opposite sides are parallel to each other. The height of the parallelogram is used to calculate its area.

diagram of a parallelogram with dimensions

Area of a parallelogram: bh

where b is the length of the base and h is the length of the height.

Area of a trapezium

The trapezium is a four-sided figure in which at least two of its sides are parallel. The parallel sides are called the bases of the trapezoid.

diagram of the dimensions of a trapezoid

Area of a trapezium: \frac{a + b}{2} h

where a and b are the lengths of both bases and h is the height of the trapezium.

Area of an ellipse

An ellipse is a figure that has an elongated circular shape.

diagram of ellipse

Area of an ellipse: πab

where a is half the length of the major axis and b is half the length of the minor axis.

FigureArea Terms
CircleA = πr²r=radius
TriangleA = ½ bhb=base, h=height
SquareA = l²l=length of one side
RectangleA = aba=width, b=base
ParallelogramA = bhb=base, h=height
TrapeziumA = \frac{a+b}{2}ha and b=bases, h=height
EllipseA = πaba=½ major axis, b=½ minor axis
Regular polygonA = ½ nlan=n° of sides, l=length of side, a=apothem

Formulas for the area of 3D figures

In the case of 3D figures, their area is called the surface area. The most important 3D figures are the cube, the rectangular prism, the cylinder, the cone, the sphere, the triangular pyramid, and the rectangular pyramid.

Area of a cube

A cube is a three-dimensional figure that has all its sides of equal length. A cube has a total of six square faces and its area is equal to the sum of all the faces.

dimensions of a cube

Area of a cube = 6a²

where a is the length of one of the sides of the cube.

Area of a rectangular prism

The rectangular prism is a 3D figure with six rectangular faces. The opposite faces of a rectangular prism are equal.

diagram of a rectangular prism with dimensions

Area of a rectangular prism = 2(lb+lh+hb)

where l is the length of the width, b is the length of the base, and h is the length of the height.

Area of a cylinder

A cylinder is a three-dimensional figure that has two circular bases, which are joined and covered by a surface.

diagram of a cylinder with radius and height

Area of a cylinder = 2πr(r+h)

where r is the radius of one of the cylinder faces, h is the height and π is a mathematical constant with a value of approximately 3.1415…

Area of a triangular pyramid

A triangular pyramid is a three-dimensional figure that has four triangular faces, which is why it is also called a tetrahedron.

diagram of a triangular pyramid with dimensions

Area of a triangular pyramid = \frac{1}{2} ba + \frac{3}{2} bh

where b is the length of one of the sides of the base, a is the height of the triangle at the base, and h is the height of one of the lateral triangles.

Area of a rectangular pyramid

A rectangular pyramid is a figure that has a rectangular base and four lateral triangular faces.

diagram of a rectangular pyramid

Area of a rectangular pyramid = ba+bh+ah

where, a and b are the lengths of the rectangular base and h is the height of a triangular side face.

Area of a sphere

A sphere is a three-dimensional figure that is perfectly round. Each point on the sphere is located at the same distance from the center.

diagram of the surface area of a sphere with radius

Area of a sphere = 4πr²

where r is the radius of the sphere and π is a numerical constant that has a value of 3.1415…

FigureArea Terms
CubeA = 6a²a=length of a side
Rectangular prismA = 2(lb+lh+hb)l=width, b=base, h=height
CylinderA = 2πr(r+h)r=radius of bases, h=height
Triangular pyramidA = \frac{1}{2}ba+\frac{3}{2}bhb=sides of base, a=height of base, h=height of lateral sides
Rectangular pyramidA = ba+bh+aha and b=sides of base, h=height of lateral sides
ConeA = πr(r+l)r=radius of base, l=slant height
SphereA = 4πr²r=radius

See also

Interested in learning more about geometric figures? Take a look at these pages:

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