Area of Isosceles Triangle

Introduction

The isosceles triangle area is the amount of surface or space enclosed between the isosceles triangle's sides. The quantity of space encircled by an isosceles triangle in a two-dimensional space is known as its area. Half of the product of the triangle's base and height is the usual formula for calculating its area. Here, to help you better comprehend this subject, a thorough explanation of the isosceles triangle area, its formula, and derivation is provided along with a few solved example problems. Isosceles triangle is one of the types of the triangle based on lengths of the sides. In this triangle any two sides or any two angles are equal. In this tutorial we understand about the area of an isosceles triangle.

Isosceles Triangles

An isosceles triangle in geometry is a triangle with two equal-length sides. It can be stated as having exactly two equal-length sides or at least two equal-length sides, with the latter definition containing the equilateral triangle as an exception. The isosceles right triangle, the golden triangle, the faces of bipyramids, and some Catalan solids are all examples of isosceles triangles.

Area of an Isosceles triangle

The whole 2-D space occupied by an isosceles triangle is referred to as its area. Based on the isosceles triangle's known components, the area of an isosceles triangle can be determined in a variety of methods. The following is a general fundamental formula for calculating the area of an isosceles triangle using height −

$$\mathrm{Area=\frac{1}{2}×base×height}$$

In the above triangle ABC, BC is a base and AD is a height.

In the above triangle ABC, BC is a base and AD is a height.

To find out value of the height we use Pythagoras theorem as follows,

$$\mathrm{DC^2+AD^2=AC^2}$$

$$\mathrm{\frac{a^2}{4}+AD^2=b^2}$$

$$\mathrm{AD^2=\frac{4b^2-a^2}{4}}$$

So,

$$\mathrm{AD=\frac{1}{2} \sqrt{4b^2-a^2 }}$$

Thus, the area of the triangle is given by,

$$\mathrm{Area=\frac{1}{2}×base×height}$$

$$\mathrm{Area=\frac{1}{2}×a×\frac{1}{2} \sqrt{4b^2-a^2 }=\frac{1}{4} a\sqrt{4b^2-a^2 }}$$

Solved Examples

1)The perimeter of an isosceles triangle is 60 cm. If one of the equal sides measures 20 cm, then find the measure of the non-equal side.

Answer:

Let a,a & b be three sides of an isosceles triangle. Then,

$$\mathrm{a+a+b=60}$$

$$\mathrm{2a+b=60}$$

$$\mathrm{2(20)+b=60}$$

$$\mathrm{b=60-40=20}$$

Thus, the length of the non-equal side is 20 cm.

2)What is the area of an isosceles triangle with a lateral a side 3 cm and base b side a 6 cm?

Answer:

Using the formula $\mathrm{Area=\frac{1}{4} a\sqrt{4b^2-a^2}}$ we get,

$$\mathrm{Area=\frac{1}{4}×3×\sqrt{4(6)^2-3^2}=\frac{3×√135}{4}=\frac{9√15}{4} \:sq. cm}$$

3)The base of an isosceles triangle is seven, while it's one of equal side is five. What is the triangle's perimeter?

Answer:

Let a,a & b be three sides of an isosceles triangle. Then,

$$\mathrm{a+a+b=P}$$

$$\mathrm{2a+b=P}$$

$$\mathrm{2(5)+7=P}$$

$$\mathrm{P=10+7=17}$$

Thus, the perimeter of an isosceles triangle is 17 cm.

4)Determine the area of an isosceles triangle if its base is 18 cm and height is 9 cm.

Answer: The area of an isosceles triangle is given by,

$$\mathrm{Area=\frac{1}{2}×base×height}$$

$$\mathrm{Area=\frac{1}{2}×18×9=9×9=81\: sq. cm}$$

5)Determine the height of an isosceles triangle whose area is 100 sq. cm and base 16 cm.

Answer: The area of an isosceles triangle is given by,

$$\mathrm{Area=\frac{1}{2}×base×height}$$

$$\mathrm{100=\frac{1}{2}×16×height}$$

$$\mathrm{height=\frac{100}{8}=\frac{25}{2}cm}$$

Thus the height of the given equilateral triangle is 12.5 cm

6)Determine the base of an isosceles triangle whose area is 100 sq. cm and height 16 cm.

Answer:

The area of an isosceles triangle is given by,

$$\mathrm{Area=\frac{1}{2}×base×height}$$

$$\mathrm{100=\frac{1}{2}×base×16}$$

$$\mathrm{base=\frac{100}{8}=\frac{25}{2}cm}$$

Thus, the base of an isosceles triangle is 12.5 cm

7)What is the area of an isosceles triangle with a 5 cm lateral a side and a 9 cm base b side?

Answer: Using the formula $\mathrm{Area=\frac{1}{4} a\sqrt{4b^2-a^2}}$ we get

$$\mathrm{Area=\frac{1}{4}×5×\sqrt{4(9)^2-3^2}=\frac{5×√315}{4} \: sq. cm}$$

8)The base of an isosceles triangle is eleven, while it's one of equal side is ten. What is the triangle's perimeter?

Answer: Let a,a & b be three sides of an isosceles triangle. Then,

$$\mathrm{a+a+b=P}$$

$$\mathrm{2a+b=P}$$

$$\mathrm{2(10)+11=P}$$

$$\mathrm{P=20+11=31}$$

Thus, the perimeter of an isosceles triangle is 31 cm.

9)The base of an isosceles triangle is twenty, while it's one of equal side is seventeen. What is the triangle's perimeter?

Answer:

Let a,a & b be three sides of an isosceles triangle. Then,

$$\mathrm{a+a+b=P}$$

$$\mathrm{2a+b=P}$$

$$\mathrm{2(17)+20=P}$$

$$\mathrm{P=34+20=54}$$

Thus, the perimeter of an isosceles triangle is 54 cm.

Conclusion

• An isosceles triangle is a triangle with two equal sides.

• Area of an isosceles triangle can be found out using the usual formula of half into base height.

• If sides of an isosceles triangle are given then we use following formula

  $$\mathrm{Area=\frac{1}{4} a\sqrt{4b^2-a^2}}$$

FAQs

1. What makes an isosceles triangle unique?

An isosceles triangle has two equal sides as well as two equal angles. The Greek words iso (same) and Skelos are the source of the name (leg). An equilateral triangle is one in which all of its sides are equal, whereas a scalene triangle is one in which none of its sides are equal.

2. What are the three characteristics of an isosceles triangle?

The characteristics of an isosceles triangle are as follows: There is agreement between two sides. The base of an isosceles triangle refers to the third side of the triangle, which is unequal to the other two sides. The two angles that are opposite the equal sides line up perfectly.

3. What about an isosceles triangle is always true?

When two sides of an isosceles triangle are equal, the angles across from those two sides likewise match up and are always equal.

4. How many different kinds of isosceles triangles exist?

The isosceles triangle is typically divided into several varieties, including the isosceles acute triangle. The right triangle isosceles. Obtuse, isosceles triangle.

5. Do all isosceles triangles look the same?

For a few reasons, all isosceles triangles are distinct from one another. The two equal sides can have the same length, but the base and base angles, as well as the measure of the angle between the two equal sides, will alter.

6. What are the isosceles triangle's five characteristics?

The properties of an isosceles triangle are as follows −

• It has two sides that are the same length.

• The opposite angles of equal sides have identical measurements.

• The perpendicular bisector of the base BC is the height from vertex A to the base BC.

• The angle bisector of the vertex angle A is the height from vertex A to base BC.

7. Can the sides of an isosceles triangle be equal?

Any two sides that are equal have equal opposite angles since every equilateral triangle is likewise an isosceles triangle.

8. What is an acute isosceles?

An acute isosceles triangle will, by definition, have no angle greater than 90 degree, at least two sides that are congruent, and at least two corresponding angles.

9. Isosceles triangles sturdy?

The equilateral triangle and the isosceles triangle are the naturally strongest triangles since they both have equal sides. The two sides that meet at a 90 degree angle in the middle are the ones whose lengths ought to be equal.