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Area of Triangle

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The triangle

Thursday, February 2nd, 2023

The Triangle

The triangle is one of the simplest polygons there is. It is a polygon enclosed by three straight segments. A triangle has three vertices, three sides, and three angles. The length of sides and the size of angles of a triangle are related. We can use these relationships to determine the lengths of sides, angles, area, and more.

An obtuse angle triangle, a right triangle, and an acute angle triangle

The Right Triangle

The right triangle is the most frequently studied type of triangle. It is at the core of trigonometric ratios and the Pythagorean theorem. Some of the common properties of right triangles are:

One angle is $90\degree$
The sum of the two remaining angles is $90\degree$
The longest side is known as the hypotenuse
The right angle is always on the opposite of the hypotenuse

Two right triangles with hypotenuse labelled

The area of a Right Triangle

The formula for the area of a right triangle is

\frac{1}{2} \times \text{base} \times \text{height}

It is actually derived from the area of a rectangle. Let’s use an example to demonstrate the relationship between the area of a rectangle and a triangle.

Example

Example 1. How to find the area of a right triangle of base 6 cm and height 4 cm

Answer:

If we enclose the right triangle in a 4x6 rectangle we get two identical right triangles, each of base 6 cm and height 4 cm.

A 4x6 rectangle divided into right triangles across its diagonal

Since the area of a rectangle is the product of its width and length, we get:

\text{Area of the Rectangle} = 6\text{cm} \times 4\text{cm}

\text{Area of the Rectangle} = 24 \text{ cm}^2

The two triangles that make up the entire rectangle should have an area of 24

\text{cm}^2

. Hence, the area of each right triangle would be half of 24

\text{cm}^2

\text{Area of one right triangle} = \frac{1}{2}\times 24

\text{Area of one right triangle} = 12 cm^2

Hence, the area of a right triangle with base 6cm and height 4cm is 12

\text {cm}^2

The area of an arbitrary right triangle will always be half the area of the rectangle that encloses it.

Area of a Non Right Triangle

Determining the area of non right triangles has significant use cases in construction, architectural design, and even estimating areas of non-regular polygons. The relationship between the area of the rectangle and triangle illustrated in the previous example holds for non-right triangles too.

Proof: Shown below is a non right triangle

Let’s enclose the triangle ABC inside a rectangle AEFC. You may notice that the triangle BCF(shaded in red) and BCD are congruent, while triangle ABE (shaded in green) and triangle ABD are congruent.

Non right triangle enclosed in rectangle

From visual inspection, we can conclude that the area of ABC is equal to the sum of the areas of triangle ABE and BCF. Hence, triangle ABC occupies an area that is half of the entire rectangle AEFC.

\text{Area of } AEFC = AC\times AE

\text{Area of } AEFC = AC\times h

Therefore,

\text{Area of triangle } ABC = \frac{1}{2}\times AC \times h

Did you notice that this is similar to the formula of the area of a right triangle? The formula for the area of a right triangle is a specific case of the area of a non right triangle.

Therefore the general formula fit for both right and non right triangles can be given as:

\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}

Question

Question 1. Find the area of the following triangle

Solution:

Step 1. Identify the base and perpendicular height.

\text{Length of base (AB) = } 7 \text{ cm}

\text{Perpendicular height} = 5 \text{ cm}

Step 2. Apply the formula for the area of a non right triangle

\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}

\text{Area} = \frac{1}{2} \times 7 \times 5

\text{Area} = 17.5 \text{ cm}^2

When the perpendicular height is unknown, the alternate formula

\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)

is beneficial.

This formula is also known as the Side-Angle-Side area formula and is useful because it applies to both right and non right triangles.

Note that when C =

90\degree

, the formula resolves to

\frac{1}{2}ab

, the formula for the area of a right triangle discussed earlier.

Let’s put the universal formula into action using an example.

Example

Example 2. Find the area of a non right triangle when the height isn’t given

Answer:

When applying the SAS area formula, identifying the parameters is essential.

a denotes the side opposite to the angle A
b denotes the side opposite angle B
c denotes the side opposite angle C

According to the definitions

\text{a} = 10.3 \text{ cm },\text{ b} = 8.4\text{ cm }, \text{c} = 7.2 \text{ cm ,} \text{ C} = 44\degree

\text{Area of triangle } ABC = \frac{1}{2} \times a \times b \times \sin(C)

\text{Area of triangle } ABC = \frac{1}{2} \times 10.3 \times 8.4 \times \sin(44\degree)

\text{Area of triangle } ABC = 29.9 \text{ cm}^2

Pythagoras theorem

Also known as the right angle formula, the Pythagoras theorem describes the mathematical relationship between the lengths of the three sides of a right triangle.

The famous Greek philosopher Pythagoras discovered this mathematical relationship, and the following shows the geometric proof he used to derive the theorem.

Geometric proof of the pythagoras theorem

The above diagram shows that the sum of the areas of the two squares of lengths AB and AC is equal to the sum of the area of the square of length equivalent to the hypotenuse of the right triangle.

For a triangle of sides a, b, and c, where c is the hypotenuse, the Pythagorean theorem formula can be given as:

a^2 + b^2 = c^2

Example

Example 3. A cable car moves from point H to I. During the journey, it descends a height of 75 m and covers a horizontal distance of 100 m. Find the length of the cable.

Answer:

Triangle IKH is a right triangle as line HK is vertical, and IK is horizontal.

To find the length of HI we can apply the Pythagorean theorem.

a^2 + b^2 = c^2

75^2 + 100^2 = c^2

5625 + 10000 = c^2

c^2 = 15625

c = \sqrt{15625}

c = 125

Hence the length of the cable must be 125 meters.

Triangle Properties Quiz

Question

Triangle Properties Quiz. Determine the area of the triangle given below

An obtuse angle triangle with certain side lengths labelled

Solution:

Here's how you'd work out the area of non right triangle ABC. The length of base AB is given as 7cm. The perpendicular height from AB to the vertex C is not given. Hence it needs to be calculated.

Step 1. Use the Pythagorean theorem to determine h

An obtuse angle triangle with perpendicular height marked

BCD is right triangle, and the sides BC and BD are:

BC = 5 \text{ cm}

BD = 10\text{ cm} - 7 \text{ cm} = 3 \text{ cm}

We can apply the Pythagorean theorem to the right triangle BCD.

h^2 + BD^2 = BC^2

h^2 = BC^2 - BD^2

h^2 = 5^2 - 3^2

h^2 = 25 - 9

h^2 = 16

h = 4 \text{ cm}

Step 2. Workout the area using the non right triangle area formula.

\text{Area} = \frac{1}{2} \times \text{base} \times \text{perpendicular height}

\text{Area} = \frac{1}{2} \times AB \times h

\text{Area} = \frac{1}{2} \times 7 \text{ cm} \times 4 \text{ cm}

\text{Area} = 14 \text{ cm}^2