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How to find area of square, rectangle, triangle, circle, cylinder, trapezoid, prism, sphere, parallelogram, cone, polygon

how to find area

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Measuring the border dimension of shapes is relatively easy. You just run a ruler or calculation around the edge. But there is more to understand when you want to know how to the find the area of a shape, whether flat or three-dimensionsal. We outline step by step how to find area of a square, rectangle, triangle, circle, semicircle, cylinder, trapezoid, prism, sphere, parallelogram, cone, polygon and more.

Geometry is not only concerned with computing itself, but also brings real objects optically into mathematics. For example, real things are shown as graphics and calculated with them. Before we get to the calculation of areas and bodies, however, let’s deal with the basic shapes. 

In mathematics lessons, students learn solutions for a range of different shapes. However, this is not that easy, depending on the shape, the area is calculated according to different formulas. We will show you how you can teach your children how to find the area of a shape. Also read: How to find interquartile range (IQR) easily: step-by-step guide

How to find the area of a square

Definition of a square

how to find area
  • the four sides are the same length
  • the four inner angles are the same size
  • it has four axes of symmetry
  • the two diagonals are the same length

In how to find area of a square: Area (A) is calculated from length times width, which is, however, the same for a square … the length is exactly equal to the width.
Formula: A = length x width

Example 1:

The length of a square is 5m, the width is 5m. What is the area of the rectangle?

Solution: A = 5m x 5m = 25 sq m

Example 2:

The area of a rectangle is 25 sq m. One side is 5 meters long. How long is the other side?

Solution: 25 sq m / 5m = 5m

How to find the area of a rectangle

how to find area

Definition of a rectangle

How do you recognize a rectangle? Well, for every rectangle, opposite sides are the same length and parallel. The two diagonals are the same length.

In how to find area of a rectangle, Area (A) is calculated from length times width, which is not the same length.
Formula: A = length x width

How to find the area of a triangle

how to find area

In how to find area of a triangle or how to find area of a right triangle:

“a” is the length of the base side of the triangle
“h” is the height of the triangle

Area: A = 0.5 · a · h
Example: a = 3cm, b = 5cm

Solution: A = 0.5 · 3cm · 5cm = 7.5cm2

How to find the area of an equilateral triangle

how to find area

An equilateral triangle has the following characteristics:

  • Three equal-length pages
  • Three axes of symmetry
  • Three angles with 60°

An equilateral triangle is centrically symmetrical, because the three axes of symmetry intersect at one point, the elevation intersection. Each axis of symmetry divides the triangle into two congruent right-angle triangles. The graphic above shows you an equilateral triangle:

Example: If you use for the length of a = 2m, you get the area A = 1.732m.

How to find the area of an isosceles triangle

An isosceles triangle is a triangle with the following characteristics:

  • Two equal-length pages ( a = b )
  • Two equal base angles ( α = β )
  • An axis ( symmetry line ) halves the base and the angle γ at the tip

The following graphic illustrates this:

how to find area

How to find the area of an irregular triangle (scalene triangle)

An irregular triangle is a triangle whose sides are all different in length. Or in other words: Pages a,b and c are of different lengths. Mathematically speaking, this looks like this: a ≠ b, a ≠ c, b ≠ c. Although everyone is surely clear what such a triangle looks like, here again a graphic.

The area of the irregular triangle (or scalene triangle) is obtained by taking half of the product of the base to the height of the triangle. Thus, the formula for the area of the scalene triangle, with a base “b” and height “h” is “(1/2) bh”.

Formula: A = [(1/2) × base × height]

How to find area of a circle

how to find area

How to find the area of a circle? To do this, let’s first look at a circle. We recognize: The circle has a center, the boundary of the circle is always the distance “r” from this center. The graphic shows above you this.

In addition to the radius, there is also the so-called diameter. The diameter is twice as large as the radius. In addition, the so-called circle number (spoken: pi) is also needed. In school, the number 3.14159 is usually used for .

Area Circle Formulas:

The two formulas for calculating the area of a circle look like this:

“A” is the area of the circle in square meters [ m2 ]
“Cul” is the circle number 3.14159
“r” is the radius of the circle in meters [ m ]
“d” is the diameter of the circle in meters [ m ]

Example 1: How to find area of a circle with the radius

The radius of a circle is 0.34m. Both above formulas are used to calculate the area of the circle.

Solution: We insert the radius into the first equation and use it to calculate the area. In order to use the second formula, the radius must first be doubled to obtain the diameter. We insert this diameter into the second equation.

How to find the area of a circle with the diameter

In order to use the formula above for how to find area of circle with diameter, the radius must first be doubled to obtain the diameter. We insert this diameter into the second equation.

how to find area

How to find area of semicircle

A semicircle is a circle is half of a circle. Therefore, the easiest way to find the area of a semicircle is to use the radius or diameter to find that area it would be if it was a full circle (see steps above), then divide that by half to give you the area of a semi circle.

How to find the area of a trapezoid

First a quick reminder: A flat figure enclosed by four lines is called a square. The trapezoid belongs to the quadrangle class and has the following properties:

  • A square with at least two parallel sides
  • The two parallel sides are called the base of the trapezoid.
  • One of these base sides (usually the longer one) is often referred to as the base of the trapezoid.
  • The height h of the trapezoid is the distance between the two parallel sides.
  • Each trapezoid has two diagonals that intersect in equal proportions.

How to find area of a trapezoid: The area A is calculated from the width times the height, where the width is calculated by adding the two parallel sides together and dividing by 2.

Formula: A = (a + c) x 0.5 x h

How to find surface area of a prism

how to find area

A prism is a geometric body that has a polygon as its base and whose side edges are parallel and of equal length. A prism is created by parallel displacement of a flat polygon along a straight line in space that is not in this plane and is therefore a special polyhedron. A distinction is often made between a straight and an oblique prism, with the graphic above showing a straight traingular prism.

How to find the area of a rectangular prism

The steps for how to find the surface area of a rectangular prism are quite simple. You find the area of each of the four sides (using the formular above for how to find the area of a rectangle), and add them together.

how to find area of a rectangular prism

How to find surface area of triangular prism

how to find area

In this section we deal with the formulas for how to find surface area of a prism. To do this, we first provide you with the formulas including the description of the variables and examples for better understanding. Let’s start with the formulas:

Where:

“V” is the volume of the prism
“A G ” is the base of the prism
“h” is the height of the prism
“A M ” is the outer surface
“U G ” is the perimeter of the base area
“O” is the surface of the prism

Example 1: how to find surface area of triangular prism

The base of the prism consists of a right triangle. The cathets are 9cm and 12cm long and the height is 20cm. The volume, the lateral surface and the surface are to be calculated.

Solution: Of course, we use the formulas for calculating the volume to calculate the volume. Since this is a right triangle, we can work with the formula for the area of ​​a triangle here.

Next we calculate the surface area. For this we need the size of the base area. We have the lengths of the cathetus, so the length of the hypotenuse is still missing. The sum of these three lengths is then the circumference of the base of the prism. This value we multiply the height by A M to obtain the surface area.

Finally, we determine the surface. We take the values ​​for A G and A M from previous calculations in order to now determine the surface.

How to find the surface area of a cylinder

how to find area

First of all, in how to find the area of a cylinder, we should briefly clarify what a cylinder is. According to the general definition, a cylinder is delimited by two parallel, flat surfaces (base and top surface) and a shell or cylinder surface that is formed by parallel straight lines. The graphic above shows you a cylinder.

How to find surface area of a cylinder: To calculate the surface of a cylinder, you need the radius “r” of the cylinder and its height “h”. The formula for calculating the cylinder surface “A” is then as follows:

A = 2 π r (r + h)

Example:

The radius of a cylinder is 2 meters, the height is 3 meters. What is the surface area of ​​the cylinder?

Solution: From the text we take the information r = 2m and h = 3m. We put these values ​​in the formula.

A = 2 3.14159 2m (2m + 3m)
A = 12,566m (5m)
A = 62,83m 2
The surface of the cylinder is 62.83m 2 .

How to find the area of the parallelogram

how to find area

Definition of the parallelogram. First a quick reminder: A flat figure enclosed by four lines is called a square. The parallelogram belongs to the class of quadrilaterals and has the following properties:

  • Two opposite sides are parallel and of the same length
  • The sum of neighboring angles is 180 °
  • The opposite angles are the same
  • The diagonals bisect each other
  • The center of symmetry is the intersection of the diagonals.

Area : The area A is calculated from the width times the height.
A = a x h a

How to find the area of a rhombus

how to find area

This is the same set of steps as for how to find the area of a parallelogram. But also you can use the diagonals.

A flat figure enclosed by four lines is called a square. The rhombus belongs to the class of squares and has the following properties:

  • A rhombus is a flat, convex square with four sides of equal length (equilateral square)
  • The opposite sides are parallel to each other
  • The two diagonals are axes of symmetry.
  • The diagonals are perpendicular to each other and bisect each other.
  • Opposite angles are the same size
  • The angle sum of all interior angles is 360 °.
  • Adjacent angles add up to 180 °.

Area : The area A is calculated as follows.
A = 0.5 * e * f
Circumference : The circumference U of the diamond is the sum of the route lengths.
U = a + b + c + d
U = 4a

How to find the area of a quadrilateral

This is the same set of steps as for how to find the area of a parallelogram, if you need to know how to find the area of a quadrilateral that is regular.

How to find the area of a polygon

how to find area

To find the area of a regular polygon, all you have to do is follow this simple formula: area = 1/2 x perimeter x apothem.[1] Here is what it means:
Perimeter = the sum of the lengths of all the sides
Apothem = a segment that joins the polygon’s center to the midpoint of any side that is perpendicular to that side.

how to find the area of an irregular shape

To find the area of irregular shapes, we can divide the irregular shape into regular polygons and then find the area of each individual polygon. Therefore, the area of the given irregular shape = Area of all the regular polygons that the irregular shape is divided into.

How to find the surface area of a sphere

Surface of a sphere Formula:

  • “O” is the surface of the sphere
  • “π” is the circle number (3.14159)
  • “r” is the radius of the sphere

Example: r = 2 cm

Solution:


How to find the surface area of a cone

how to find area

The first question that arises is what a circular cone actually is? Well, a straight circular cone has the radius r as its base, the tip is at height h vertically above the center of the circle. The graphic above shows you what a straight circular cone looks like and also introduces the corresponding variables for the formulas.

To the formulas for the straight circular cone:

Where:

  • “V” is the volume of the circular cone
  • “π” is the circle number, about 3.14159
  • “r” is the radius of the base
  • “h” is the height of the circular cone
  • “A M ” is the outer surface
  • “A O ” the surface
  • “s” is the length of the page

Example 2: Calculate the surface area of a cone

A circular cone has a radius of 40cm and a height of 60cm. We first calculate the length of the “s” side, as we need this information for the further calculation. Then we just have to insert the values ​​in the other formulas and calculate everything.

How to find area of a sector or segment of a circle

how to find area

In geometry, a segment of a circle is a partial area of ​​a circular area that is delimited by an arc and a chord. This can also be seen in the graphic above. The formulas including the example also refer to this graphic:

Where :

  • “α” the angle at the center point (see graphic)
  • “b” is the length of the arc from A to B.
  • “h” is the height of the segment
  • “r” is the radius of the circle
  • “s” is the length of the circular tendon

In addition, there are a few others to note. “A” is the area of ​​the area drawn in green. The “M” stands for the center of the circle. For a better overview, the points “A” and “B” have also been introduced, which denote the ends of the circular arc.

Circle segment formulas and example

Next, we look at the formulas for the segment of a circle or segment of a circle. The following formulas can be used to make calculations for this.

Example: The area of ​​a circle segment is to be calculated. The following information applies: h = 2cm, s = 6cm and b = 9cm. How big is the area of ​​the circle segment? The solution is:

How to find surface area of a square pyramid

how to find area

There are several methods for how to find the surface area of a square pyramid. Here are some formulas for this. What the individual variables stand for can be seen in the graphic. The missing information is A O = surface of the pyramid, A G = base of the pyramid, A M = outer surface of the pyramid.

Tip: You should use all information in meters, then you will get a result in square meters.

Example: The base edge has a length of 4 meters. The side height is 5 meters. How big is the surface of the pyramid?

Solution: From the text we take the information a = 4m and h s = 5m. We put this information into the formula:

The pyramid thus has a surface of 56m 2 .

Conclusion

Geometry is not necessarily a student’s favorite subject, but the knowledge learned in school in particular will be queried again and again in later years and will be needed in mathematics lessons.

Some children thrive when they are finally allowed to use a compass or when they can show their good sense of space while drawing shapes and figures. Most of them, however, groan in annoyance when calculating angles or determining cube surface or volume.

Geometry as a sub-area of ​​mathematics is already part of the classroom in the first grade of primary school. Rightly so, because in many areas of everyday life, geometry helps to answer questions and solve problems. The technology, which is at least 4000 years old, was used for land surveying in ancient Egypt, for example. And even today, many professions, from fashion designers to architects, need basic geometric knowledge.

As the basics of geometry, you will get to know:

  • Get to know and discover shapes and bodies
  • Recognize bodies from different perspectives
  • Find and draw right angles and perpendiculars
  • Recognize cube networks
  • Break apart and add triangles and squares
  • Find and draw in rotational symmetry axes and mirror axes
  • Understand cube networks
  • Complete the pattern
  • Draw circles (compasses)
  • Use of triangles, rulers and compasses
  • Increase and decrease scales
  • Develop an idea of ​​the area
Important, geometric definitions
termdefinition
PointA point is an object with no extent.
routeA line is the shortest possible connection between two points.
JustA straight line is a straight line with no start and no end point.
beamA ray is a straight line that has a start point but no end point.
parallelStraight lines are parallel if they do not intersect at any point.
Complex, geometric shapes
termdefinition
parallelogramA parallelogram is a square with opposite sides of the same length.
TrapezoidA trapezoid is a square that has a pair of parallel opposite sides.
cylinderA cylinder is one of two parallel, plane, congruent
RhombusThe rhombus or diamond is a parallelogram with 4 sides of equal length. 
Geometric solids
termdefinition
coneA cone is created when all points of a bounded circular area lying in a plane are connected in a straight line to a point outside the plane.
cylinderA cylinder is a body delimited by two parallel, flat, congruent surfaces and a jacket or cylinder surface, the jacket surface being formed by parallel straight lines.
sphereThe sphere is the set of all points whose distance from the center is less than or equal to the radius r.