Loading [MathJax]/jax/output/HTML-CSS/jax.js

Perimeter and Area of a Triangle

Here we will discuss about the perimeter and area of a triangle and some of its geometrical properties.

Perimeter, Area and Altitude of a Triangle:

Perimeter, Area and Altitude of a Triangle

Perimeter of a triangle (P) = Sum of the sides = a + b + c

Semiperimeter of a triangle (s) = 12(a + b + c)

Area of a triangle (A) = 12 × base × altitude = 12ah

Here any side can be taken as base; the length of the perpendicular from the corresponding vertex to this side is the altitude.

Area = s(s - a)(s - b)(s - c)  (Heron’s formula)

Altitude (h) = area12×base = 2a


Solved Example on Finding the Perimeter, Semiperimeter and Area

 of a Triangle: 

The sides of a triangle are 4 cm, 5 cm and 7 cm. Find its perimeter, semiperimeter and area.

Solution:

Perimeter of a triangle (P) = Sum of the sides

                                      = a + b + c

                                      = 4 cm + 5 cm + 7 cm

                                      = (4 + 5 + 7) cm

                                      = 16 cm


Semiperimeter of a triangle (s) = 12(a + b + c)

                                             = 12(4 cm + 5 cm + 7 cm)

                                             = 12(4 + 5 + 7) cm

                                             = 12 × 16 cm

                                             = 8 cm

Area of a triangle = s(s - a)(s - b)(s - c) 

                          = 8(8 - 4)(8 - 5)(8 - 7) cm2

                          = 8 × 4 × 3 × 1 cm2

                          = 96 cm2

                          = 16×6 cm2

                          = 46 cm2

                          = 4 × 2.45 cm2

                          = 9.8 cm2


Perimeter, Area and Altitude of an Equilateral Triangle:

Perimeter, Area and Altitude of an Equilateral Triangle

Perimeter of an equilateral triangle (P) = 3 × side = 3a

Area of an equilateral triangle (A) = 34 × (side)2 = 34 a2

Altitude of an equilateral triangle (h) = 34 a


Trigonometric formula for area of a triangle:

Trigonometric Formula for Area of a Triangle

Area of ∆ABC = 12 × ca sin B

                    = 12 × ab sin C

                    = 12 × bc sin A

(since, ∆ = 12 ah = 12 ca ∙ hc = 12 ca sin B, etc.)


Solved Example on Finding the Area of a Triangle: 

In a ∆ABC, BC = 6 cm, AB = 4 cm and ∠ABC = 60°. Find its area.

Solution:

Area of ∆ABC = 12 ac sin B = 12 × 6 × 4 sin 60° cm2

                    = 12 × 6 × 4 × 32 cm2

                    = 6√3 cm2

                    = 6 × 1.73 cm2

                    = 10.38 cm2


Some geometrical properties of an isosceles triangle:

Geometrical Properties of an Isosceles Triangle

In the isosceles ∆PQR, PQ = PR, QR is the base, and PT is the altitude.

Then, ∠PTR = 90°, QT = TR, PT2 + TR2 = PR2 (by Pythagoras’ theorem)

 ∠PQR = ∠PRQ, ∠QPT = ∠RPT.


Some geometrical properties of a right-angled triangle:

In the right-angled ∆PQR, ∠PQR = 90°; PQ, QR are the sides (forming the right angle) and PR is the hypotenuse.

Geometrical Properties of a Right-angled Triangle

Then, PQ ⊥ QR (therefore, if QR is the base, PQ is the altitude).

PQ2 + QR2 = PR2 (by Pythagoras’ theorem)

Area of the ∆PQR = 12 ∙ PQ ∙ QR

⟹ PQ ∙ QR = 2 × area of the ∆PQR.

Again, area of the ∆PQR = 12 ∙ QT ∙ PR

⟹ QT ∙ PR = 2 × area of the ∆PQR.

Therefore, PQ ∙ QR = QT ∙ PR = 2 × Area of the ∆PQR.


Solved Examples on Perimeter and Area of a Triangle:

1. Find the perimeter of an equilateral triangle whose area is equal to that of a triangle with sides 21 cm, 16 cm and 13 cm.

Solution:

Let a side of the equilateral triangle = x.

Then, its area = 34 x2

Now, the area of the other triangle = s(s - a)(s - b)(s - c) 

Here, s = 12 (a + b + c)

           = 12 (21 + 16 + 13) cm

           = 12 50 cm

           = 25 cm

Therefore, area of the other triangle = 25(25 - 21)(25 - 16)(25 - 13) cm2

                                                     = 25 ∙ 4 ∙ 9 ∙ 12 cm2

                                                     = 603 cm2

According to the question, 34 x2 = 603 cm2

⟹ x2 = 240 cm2

Therefore, x = 4√15 cm


2. PQR is an isosceles triangle whose equal sides PQ and PR are 10 cm each, and the base QR measures 8 cm. PM is the perpendicular from P to QR and X is a point on PM such that ∠QXR = 90°. Find the area of the shaded portion.

Solved Examples on Perimeter and Area of a Triangle

Solution:

Since PQR is an isosceles triangle and PM ⊥ QR, QR is bisected at M.

Therefore, QM = MR = 12 QR = 12 × 8 cm = 4 cm

Now, PQ2 = PM2 + QM2 (by Pythagoras’ theorem)

Therefore, 102 cm2 = PM2 + 42 cm2

or, PM2 = 102 cm2 - 42 cm2

                       = 100 cm2 - 16 cm2

                       = (100 - 16) cm2

                       = 84 cm2

Therefore, PM2 = 2√21 cm

Therefore, area of the ∆PQR = 12 × base × altitude

                                         = 12 × QR × PM

                                         = (12 × 8 × 2√21) cm2

                                         = 8√21) cm2

From geometry, ∆XMQ ≅ ∆XMR (SAS criterion)

We get, XQ =XR = a (say)

Therefore, from the right-angled ∆QXR, a2 + a2 = QR2

or, 2a2 = 82 cm2

or, 2a2 = 64 cm2

or, a2 = 32 cm2

Therefore, a = 4√2 cm

Again, area of the ∆XQR = 12 × XQ × XR

                                    = 12 × a × a

                                    = 12 × 4√2 cm  × 4√2 cm

                                    = 12 × (4√2)2 cm2

                                    = 12 × 32 cm2  

                                    = 16 cm2  

Therefore, area of the shaded portion = area of the ∆PQR - area of the ∆XQR

                                                      = (8√21) cm2 - 16 cm2

                                                      = (8√21 - 16) cm2  

                                                      = 8(√21 - 2) cm2  

                                                      = 8 × 2.58 cm2  

                                                      = 20.64 cm2  






9th Grade Math

From Perimeter and Area of a Triangle to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More

  2. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 15, 25 02:46 AM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  3. 5th Grade Quadrilaterals | Square | Rectangle | Parallelogram |Rhombus

    Jul 15, 25 02:01 AM

    Square
    Quadrilaterals are known as four sided polygon.What is a quadrilateral? A closed figure made of our line segments is called a quadrilateral. For example:

    Read More

  4. 5th Grade Geometry Practice Test | Angle | Triangle | Circle |Free Ans

    Jul 14, 25 01:53 AM

    Name the Angles
    In 5th grade geometry practice test you will get different types of practice questions on lines, types of angle, triangles, properties of triangles, classification of triangles, construction of triang…

    Read More

  5. 5th Grade Circle Worksheet | Free Worksheet with Answer |Practice Math

    Jul 11, 25 02:14 PM

    Radii of the circRadii, Chords, Diameters, Semi-circles
    In 5th Grade Circle Worksheet you will get different types of questions on parts of a circle, relation between radius and diameter, interior of a circle, exterior of a circle and construction of circl…

    Read More