Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm

Find the area of a triangle two sides of which are 18 cm and 10 cm and the perimeter is 42 cm

To find the area of a triangle when two sides and the perimeter are given, we can use Heron's formula. This formula is particularly useful when we do not have the height of the triangle but know the lengths of all three sides.

Two sides of the triangle: a = 18 cm and b = 10 cm

Perimeter of the triangle: 42 cm

The perimeter of the triangle is the sum of all its sides. Therefore, we can find the third side c as follows:

42 = 18+10+c

42 = 28+c

c = 42−28

c = 14 cm

Calculating the Semi-Perimeter:

The semi-perimeter s is half the perimeter of the triangle:

s = (a+b+c​)/2 = 42/2 = 21 cm

Using Heron's Formula:

Heron's formula for the area of a triangle is

Area = √s(s - a)(s - b)(s - c)

Substituting the values into the formula

Area = √21(21 - 18)(21 - 10)(21 - 14)

Area = √21x3x11x7

The simplified form of the area is

Area = 21√11 cm²

What is the Area of a Triangle?

The area of a triangle is a fundamental concept in geometry. It represents the amount of space enclosed within the triangle's boundaries. Understanding how to calculate the area of a triangle is essential for various applications in mathematics.

The simplest formula for the area of a triangle is Area = 1/2 x base x height

This formula applies when you know the base of the triangle and its corresponding height (the perpendicular distance from the base to the opposite vertex).

Trigonometric Formula:

Another method involves using trigonometry, particularly when you know two sides and the included angle.

The formula is Area = 1/2​ x a x b x sin(C)

where a and b are two sides, and C is the angle between them.

Heron's Formula:

Heron's Formula is useful when you know the lengths of all three sides of the triangle but not the height. It states that the area can be calculated using the semi-perimeter (half the perimeter) of the triangle.

The formula is Area = √s(s - a)(s - b)(s - c)

where s is the semi-perimeter, and a, b, and c are the lengths of the sides.