#### What is a triangle?

A triangle or triangle is a basic type of geometry: a flat two-dimensional figure whose vertices are three noncollinear points and whose sides are three line segments connecting the vertices. A triangle is the polygon with the least number of sides (3 sides). The triangle is always a simple polygon and always a convex polygon (the interior angles are always less than 180o).

#### Types of triangles

Common triangle: is the most basic triangle, with different lengths of sides, different interior angles. Regular triangles can also include special cases of triangles.

Isosceles triangle: is a triangle with two equal sides, these two sides are called lateral sides. The vertex of an isosceles triangle is the intersection of the two sides. The angle formed by the vertex is called the vertex angle, the other two angles are called the base angle. The property of an isosceles triangle is that the two base angles are congruent.

You are viewing: Formula to calculate the area of a triangle

Equilateral triangle: is a special case of an isosceles triangle with all three equal sides. The property of an equilateral triangle is that all three angles are equal and equal to 60 ..

Right triangle: is a triangle with an angle equal to 90 . (is a right angle).

Obtuse triangle: is a triangle with a larger interior angle greater than 90(an obtuse angle) or has an exterior angle less than 90 (an acute angle).

An acute triangle is a triangle with all three interior angles less than 90 . (three acute angles) or all exterior angles greater than 90 (six obtuse corners).

Isosceles right triangle: both a right triangle and an isosceles triangle.

## Formula to calculate the area of a triangle: square, normal, isosceles, equilateral

### Calculate the area of an ordinary triangle

An ordinary triangle is a triangle with three different lengths of sides and unequal measures of the three angles.

Common triangle can include other special cases such as isosceles triangle, right triangle, equilateral triangle. Therefore, the same formulas below can be applied to calculate the area of many different triangles.

Calculate the area when the length of the altitude is known

The area of the triangle is ½ the product of the altitude from the vertex multiplied by the opposite side of that vertex.

Triangle ABC has three sides a, b, c, h_{a} is the altitude from vertex A as shown:

#### General Formula

The area of the triangle is ½ product of the height from the vertex to the length of the opposite side of that vertex.

#### Find the area of a triangle when one angle . is known

Example: Calculate the area of triangle ABC whose base length is 32cm and height is 22cm.

In there:

a, b, c are the lengths of the sides of the triangle, respectively.

The area of the triangle is ½ the product of the two adjacent sides and the sine of the angle joined by those two sides in the triangle.

Example: Given a triangle ABC with angle B equal to 60 degrees, side BC = 7, side AB = 5. Calculate the area of triangle ABC?

#### Calculate the area of a triangle with three sides known using Heron’s formula.

Using the proven Heron formula:

In there:

a, b, c are the lengths of the sides of the triangle, respectively.

p: Half perimeter of triangle, equal to ½ sum of the sides of a triangle.

Where p is the half-perimeter of the triangle:

It can be rewritten with the formula:

Example: Calculate the area of a triangle with side length AB = 8, AC = 7, CB = 9

#### Find the area equal to the radius of the circumcircle of the triangle (R)

In there:

a, b, c are the lengths of the sides of the triangle, respectively.

R: Radius of the circumcircle.

Let R be the radius of the circumcircle of triangle ABC.

We have:

Different ways:

Note: It is necessary to prove that R is the radius of the circumcircle of the triangle.

Example: Given a triangle ABC, the lengths of the sides a = 6, b = 7, c = 5, R = 3 (R is the radius of the circumcircle of triangle ABC).

#### Find the area equal to the radius of the circle inscribed in the triangle (r)

In there:

p: Half perimeter of triangle.

r: Radius of the inscribed circle.

Let r be the radius of the incircle of triangle ABC and p the half-perimeter of the triangle p=(a+b+c)/2.

Example: Calculate the area of triangle ABC knowing the lengths of sides AB = 20, AC = 21, BC = 15, r = 5 (r is the radius of the incircle of triangle ABC).

### Formulas for calculating the area of a triangle in space

In the Oxy plane, call the coordinates of the vertices of triangle ABC respectively: A(xA,yA),B(xB,yB),C(xC,yC), we can use the following formulas to calculate triangle area

In the Oxy plane, call the coordinates of the vertices of the triangle ABC as:

Applied in space, with the concept of a directed product of two vectors. We have:

For example: In Oxyz space for 3 points A(1;2;1), B(2;-1;3), C(5;2;-3). Calculate the area of triangle ABC.

Example exercises

* Calculate the area of a triangle with

a, Bottom length is 15cm and height is 12cm

b, Bottom length is 6m and height is 4.5m

The answer:

a) The area of the triangle is:

(15 x 12) : 2 = 90 (cm2)

Answer: 90cm2

b, The area of the triangle is:

(6 x 4.5) : 2 = 13.5 (m2)

Answer: 13.5m2

* Note: In case you do not give the bottom edge or height, but give the area and the remaining edge, please apply the formula deduced above to calculate.

Some notes when calculating the area of the triangle.

– With a triangle containing a flat angle, the height is outside the triangle, then the length of the side to calculate the main area is equal to the length of the side inside the triangle.

– When calculating the area of a triangle, which height corresponds to that base.

– If two triangles have the same height or equal height -> the area of the two triangles is proportional to the 2 bases and vice versa if the two triangles have the same base (or two equal bases) -> the area of the triangle proportional to the 2 corresponding altitudes.

## Calculate the area of an isosceles triangle

An isosceles triangle is a triangle with two equal sides and the same base angle measures.

An isosceles triangle ABC has three sides, a is the length of the base, b is the lengths of the two sides, h_{a} is the altitude from vertex A as shown:

Applying the formula for calculating the normal area, we have the formula for calculating the area of an isosceles triangle:

Example exercises

* Calculate the area of an isosceles triangle with:

a, The length of the base side is 6cm and the height is 7cm

b, The length of the bottom side is 5m and the height is 3.2m

The answer:

a) The area of the triangle is:

(6 x 7) : 2 = 21 (cm2)

Answer: 21cm2

b, The area of the triangle is:

(5 x 3.2) : 2 = 8 (m2)

Answer: 8m2

### Calculate the area of an equilateral triangle

An equilateral triangle is a triangle with three equal sides and the same angle measures 60 degrees.

An equilateral triangle ABC has three equal sides, a is the lengths of the sides as shown in the figure:

Applying Heron’s theorem to deduce, we have the formula to calculate the area of an equilateral triangle:

In there:

a: The lengths of the sides of an equilateral triangle.

The example below will help you better understand the formula for calculating the area of an equilateral triangle above.

Example: Calculate the area of an equilateral triangle ABC with side 10.

Example exercises

* Calculate the area of an equilateral triangle:

a, The length of one side of the triangle is 6cm and the height is 10cm

b, The length of one side of the triangle is 4cm and the height is 5cm

The answer

a) The area of the triangle is:

(6 x 10) : 2 = 30 (cm2)

Answer: 30cm2

b, The area of the triangle is:

(4 x 5) : 2 = 10 (cm2)

Answer: 10cm2

### Calculate the area of a right triangle

A right triangle is a triangle with an angle equal to 90 degrees (right angle).

#### Formula for calculating the area of a right triangle

For example, triangle ABC is right-angled at A. Applying the formula for calculating the area of an ordinary triangle, we have:

In there:

A, B, C: The vertices of the triangle.

a, b, c: denote the lengths of the sides BC, AC, AB, respectively.

ha: Altitude and descent from the corresponding vertex A.

S: Area of the triangle.

Triangle ABC, right angled at B, a, b are the lengths of the two sides of the right angle:

Apply the normal area formula for the area of a right triangle with the height of one of the two sides of the right angle and the base of the other side.

Formula to calculate the area of a right triangle:

Example: Calculate the area of triangle ABC whose base length is 32cm and height is 22cm.

Example exercises

* Calculate the area of a right triangle with:

a, The two sides of the right angle are 3cm and 4cm . respectively

b, Two sides of right angle are 6m and 8m . respectively

The answer:

a) The area of the triangle is:

(3 x 4) : 2 = 6 (cm2)

Answer: 6cm2

b, The area of the triangle is:

(6 x 8) : 2 = 24 (m2)

Answer: 24m2

Similarly, if the data asks back about how to calculate the length, you can use the deduced formula above.

### Calculate the area of an isosceles triangle

Triangle ABC is right-angled at A, a is the lengths of the two sides of the right angle:

Applying the formula for calculating the area of a right triangle for the area of an isosceles right triangle with the same height and base, we have the formula:

#### Self-practice on triangles grade 5

Exercise 1: Calculate the area of triangle MDC (Figure below). We know rectangle ABCD has AB = 20 cm, BC = 15 cm.

Exercise 2: Calculate the height AH of triangle ABC, right angled at A. Know: AB = 60 cm; AC = 80 cm ; BC = 100 cm.

Exercise 3: A triangle has a base 16cm long and a height equal to 3/4 of the base length. Calculate the area of the triangle

Exercise 4: A triangular piece of rock has an area of 288m2, a base side is 32m. In order to increase the area of the sand by 72m2, the given bottom edge must be increased by how many meters?

Lesson 5: A triangular scarf has a base of 5.6 dm and a height of 20 cm. Calculate the area of that scarf.

Lesson 6: A triangular garden has an area of 384m2, a height of 24m. What is the base side of the triangle?

Lesson 7: A triangular courtyard has a base of 36m and is 3 times its height. Calculate the area of that triangular yard?

Exercise 8: Let ABC be a right triangle (angle A is a right angle). The length of side AC is 12dm and the length of side AB is 90cm. Calculate the area of triangle ABC?

Exercise 9: Let ABC be a right triangle at A. Know AC = 2,2dm, AB = 50cm. Calculate the area of triangle ABC?

Lesson 10: The triangle MNP has height MH = 25cm and has an area of 2dm2. Calculate the base length NP of that triangle?

Lesson 11: A strange restaurant has the shape of a triangle with the sum of the bottom side and the height of 24m, the bottom side equal to 1515 of the height. Calculate the area of that restaurant?

Exercise 12: Let ABC be a triangle with base BC = 2cm. How much longer does BC have to be extended to get triangle ABD with an area and a half times the area of triangle ABC?

Lesson 13: A triangle whose base side is 2/3 of its height. If the base side is extended by 30dm, the area of the triangle increases by 27m2. Calculate the area of that triangle?

Lesson 14: A triangle whose base is 7/4 of its height. If the base side is lengthened by 5m, the area of the triangle increases by 30m2. Calculate the area of that triangle?

Exercise 15: Given a right triangle ABC at A. If AC is extended towards C a segment CD is 8cm long, then triangle ABC becomes isosceles right triangle ABD and the area increases by 144cm2. Calculate the area of right triangle ABC?

#### Advanced Triangle Exercises

Exercise 1: Let ABC be a right-angled triangle at A with perimeter 72cm. The length of side AB is 3/4 of the length of side AC, and the length of side AC is 4/5 of the length of side BC. Calculate the area of triangle ABC

Exercise 2: In triangle ABC, know M and N are the midpoints of sides AB and AC respectively. Find the area of triangle ABC if the area of triangle AMN is 5cm2

Exercise 3: Let ABCD be a square with AB = 6cm, M is the midpoint of BC, DN = 1/2NC. Calculate the area of triangle AMN.

Lesson 4: Let the triangle MNP. Let K be the midpoint of the side NP, and I the midpoint of the side MP. Knowing the area of triangle IKP is 3.5cm2. Calculate the area of triangle MNP

Exercise 5: Let ABC be a triangle with side AB 20cm long and side AC 25cm long. On side AB take point D 15cm from A, on side AC take point E 20cm away from point A. Connect D to E to get a triangle ADE with area 45cm2. Calculate the area of triangle ABC

Lesson 6: Let ABC be a triangle. Points D, E, G are the midpoints of sides AB, BC and AC respectively. Calculate the area of triangle DEG, knowing the area of triangle ABC is 100m2

Lesson 7: (Examination to 6 Archimedes Academy schools 2019 – 2020 – batch 2)

Given the triangle with the proportions as shown.

Knowing S3−S1=84cm2. Calculate S4-S2

Lesson 8: (Examination to 6 schools in Hanoi Amsterdam 2010 – 2011)

Let ABC be a triangle with an area of 180 cm2. Know AB = 3 x BM; AN = NP=PC; QB=QC. Calculate the area of triangle MNPQ? (see drawing)

Lesson 9: (Examination to 6 schools in Hanoi Amsterdam 2006 – 2007)

Let ABC be a triangle with an area of 18cm2. Know DA = 2 x DB ; EC = 3 x EA ; MC = MB (figure). Calculate the total area of two triangles MDB and MCE ?

Lesson 10: (Examination to 6 schools in Hanoi and Amsterdam from 2004 to 2005)

In the figure below, there are NA = 2 x NB; MC = 2 x MB and the area of triangle OAN is 8cm2. Calculate the area of BNOM?

Posted by: Soc Trang High School

Category: Education

Copyright of the article belongs to Soc Trang City High School. Any copying is fraud! Shared source: Soc Trang High School (thcsquynhxuan.edu.vn)

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