Area of Equilateral Triangle, Perimeter & Altitude Formulas

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An equilateral triangle is a triangle with equal sides and equal angles. All the internal angles are of the measure 60º. The word ‘equi’ comes from Latin and means “equal” or “the same”.

equilateral triangle

As shown in the diagram, Δ ABC is an equilateral triangle with sides of length a and an altitude of length h. This triangle will be used as reference for all the formulas of the equilateral triangle mentioned here.

Key Takeaways: Equilateral Triangle, Area of Equilateral Triangle, Perimeter of Equilateral Triangle, Length of Altitude of Equilateral Triangle, Semi-perimeter of Equilateral Triangle


Perimeter of Equilateral Triangle

Perimeter of an Equilateral Triangle = sum of the length of all sides = a + a + a = 3a

Perimeter of an Equilateral Triangle = 3a


Semi-Perimeter of an Equilateral Triangle

Semi-perimeter of an Equilateral Triangle 

= \(\frac{\text{ Perimeter of an Equilateral Triangle}}{2} = \frac{3a}{2}\)


Altitude of Equilateral Triangle

Length of the altitude of an equilateral triangle = \(\frac{\sqrt{3}a}{2}\)

Altitude of Equilateral Triangle


Area of Equilateral Triangle

Area of a triangle A = \(\frac{1}{2}\)bh

For an equilateral triangle, b = a and h = \(\frac{\sqrt{3}a}{2}\)

So, the area of an equilateral triangle = \(\frac{1}{2}a \times \frac{\sqrt{3}a}{2} \times \frac{\sqrt{3}}{4} a^2\)

Area of an Equilateral Triangle = \(\frac{\sqrt{3}}{4}a^2\)

Area of Equilateral Triangle


Things To Remember

  • In an equilateral triangle, all sides and all angles are equal, and the measure of each internal angle is 60º.
  • Perimeter of an equilateral triangle = 3a
  • The semi-perimeter of an equilateral triangle is equal to half its perimeter.
  • Length of the altitude of an equilateral triangle = \(\frac{\sqrt{3}a}{2}\)
  • Area of an equilateral triangle =\(\frac{\sqrt{3}}{4}a^2\)

Sample Questions

Ques: The length of an equilateral triangle is 12 cm. Calculate its area, perimeter, semi-perimeter, and length of altitude. (5 marks)

Ans: a = 12 cm

Area

Area of an Equilateral Triangle = \(\frac{\sqrt{3}}{4}a^2\)

Area = \(\frac{\sqrt{3}}{4}(12)^2\)=\(\sqrt{3} \times12 \times 3\) = \(36 \sqrt{3}\) cm2 

Perimeter

Perimeter of an Equilateral Triangle = 3a

Perimeter = 3 × 12 = 36 cm

Semi-perimeter

Semi-perimeter of an Equilateral Triangle =

= \(\frac{\text{ Perimeter of an Equilateral Triangle}}{2} = \frac{36}{2}\)= 18 cm 

Altitude

Length of the altitude of an equilateral triangle = \(\frac{\sqrt{3}a}{2}\)

Altitude =\(\frac{\sqrt{3} \times 12}{2} = 6 \sqrt{3}\)

The area, perimeter, semi-perimeter, and length of the altitude of this triangle are \(36 \sqrt{3}\) cm2, 36 cm, 18 cm, and \( 6 \sqrt{3}\) cm, respectively.

Ques: If the perimeter of an equilateral triangle is 99 cm, then find the length of its sides. (3 marks)

Ans: Perimeter = 99 cm

Perimeter of an equilateral triangle = 3a

99 = 3a

a = 33 cm

So, the length of each side of this equilateral triangle is 33 cm.

Ques: Calculate the area of an equilateral triangular park whose each side measures 200 m. (3 marks)

Ans: a = 200 m

Area of an Equilateral Triangle = \(\frac{\sqrt{3}}{4}a^2\)

Area of triangular park = \(\frac{\sqrt{3}}{4}(200)^2\) = \(\sqrt{3} \times12 \times 3\) = \(10000 \sqrt{3}\) m2 

So, the area of this equilateral triangular park is \(10000 \sqrt{3}\)m2.

Ques: If the semi-perimeter of an equilateral triangle is 15 cm, then calculate its perimeter, area, and length of altitude. (5 marks)

Ans: semi-perimeter = 15 cm

Perimeter

Perimeter of equilateral triangle = 2 × semi-perimeter = 2 × 15 = 30 cm

Perimeter of equilateral triangle = 3a

30 = 3a

a = 10 cm

Area

Area of an Equilateral Triangle = \(\frac{\sqrt{3}}{4}a^2\)

Area =  \(\frac{\sqrt{3}}{4}(10 )^2\) = \(\sqrt{3} \times5 \times 5\) = \(25 \sqrt{3}\) m2 

Altitude

Length of the altitude of an equilateral triangle = \(\frac{\sqrt{3}}{4}a^2\)

Altitude= \(\frac{\sqrt{3} \times 10}{2} = 5 \sqrt{3}\)

The perimeter, area, and length of altitude of this equilateral triangle are 30 cm, \(25 \sqrt{3}\) cm2, and \(5 \sqrt{3}\) cm, respectively.

Ques: Calculate the area of an equilateral triangle whose perimeter is 45 cm. (3 marks)

Ans: perimeter = 45 cm

Perimeter of an equilateral triangle = 3a

45 = 3a

a = 15 cm

Area of an Equilateral Triangle = \(\frac{\sqrt{3}}{4}a^2\)

Ques: Calculate the semi-perimeter of an equilateral triangle with an altitude of 20 units. (4 marks)

Ans: h = 20 units

Length of the altitude of an equilateral triangle =\(\frac{\sqrt{3}a}{2}\)

Semi-perimeter of an Equilateral Triangle = \(\frac{\sqrt{3}a}{2}\)

Semi-perimeter = 

So, the semi-perimeter of this triangle is \(20 \sqrt{3}\) units.

Ques: Calculate the area of a circumscribed circle of an equilateral triangle with a length of each side of 5 cm. (3 marks)

Ans: a = 5 cm

The formula for calculating the radius of a circumscribed circle of an equilateral triangle is given by r = \(\frac{\sqrt{3}a}{3}\)

Area of a circle, 

So, the area of this circumscribed circle of an equilateral triangle is \(\frac{25 \pi}{3}\) cm2.

CBSE X Related Questions

  • 1.
    AB and CD are diameters of a circle with centre O and radius 7 cm. If \(\angle BOD = 30^\circ\), then find the area and perimeter of the shaded region.


      • 2.
        If the zeroes of the polynomial $ax^2 + bx + \dfrac{2a}{b}$ are reciprocal of each other, then the value of $b$ is

          • $\dfrac{1}{2}$
          • 2
          • -2
          • $-\dfrac{1}{2}$

        • 3.
          Let $p$, $q$ and $r$ be three distinct prime numbers. Check whether $pqr + q$ is a composite number or not. Further, give an example for three distinct primes $p$, $q$, $r$ such that
          (i) $pqr + 1$ is a composite number
          (ii) $pqr + 1$ is a prime number


            • 4.

              In the adjoining figure, TS is a tangent to a circle with centre O. The value of $2x^\circ$ is

                • 22.5
                • 45
                • 67.5
                • 90

              • 5.

                Given that $\sin \theta + \cos \theta = x$, prove that $\sin^4 \theta + \cos^4 \theta = \dfrac{2 - (x^2 - 1)^2}{2}$.


                  • 6.
                    In a right triangle ABC, right-angled at A, if $\sin B = \dfrac{1}{4}$, then the value of $\sec B$ is

                      • 4
                      • $\dfrac{\sqrt{15}}{4}$
                      • $\sqrt{15}$
                      • $\dfrac{4}{\sqrt{15}}$

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