Area of Isosceles Triangles

🏆Practice area of a triangle

Formula to calculate the area of an isosceles triangle

A1 - Area of the isosceles triangle

Height of the base × Base2=A \frac{Height~of~the~base~\times ~Base}{2}=A

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Test yourself on area of a triangle!

Calculate the area of the right triangle below:

101010666888AAACCCBBB

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Area of Isosceles Triangles

Calculating the area of an isosceles triangle is very simple, easy, and even identical to the calculation we do to find out the area of other types of triangles. Therefore, if you happen to get a question about calculating the area of isosceles triangles on the exam, I assure you that a small smile will appear on your face.


How is the area of an isosceles triangle calculated?

We will multiply the base by the height and divide by two.

A1 - Area of the isosceles triangle

Remember!

The main property of the isosceles triangle is that the median of the base, the bisector, and the height are the same, that is, they coincide. Therefore, even if the question only names the median of the base or the bisector, you can immediately deduce that it is also the height of the triangle and use it to calculate its area.

Observe the theorem holds true only with the height, the median of the base, and the bisector!

You didn't think we were going to send you off without any exercises on the topic, did you? Time to practice!

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Let's start with a classic exercise

Here you have an isosceles triangle ABCABC

A4 - Practice calculating the area of the isosceles triangle

Given that:
ab=acab=ac
ADAD -

Height
AD=4AD = 4
CB=6CB=6

What is the area of the triangle?

Solution: We will proceed according to the formula - the height AD=4AD = 4
multiply by the base CB=6CB = 6
and divide the received product by 22
We will obtain:
4×62=12 \frac{4\times6}{2}=12
The area of the triangle ABCABC is 1212 cm2.


Now let's move on to an exercise that aims to be a bit more sophisticated:

You have the isosceles triangle FDCFDC

A3 - Exercise on calculating the area of the isosceles triangle

Given that:
FC=FDFC=FD
CG=4CG= 4
FG=5FG = 5 The median of the base

Calculate the area

Solution: Let's remember that, in an isosceles triangle, the median of the base is also the height, therefore, we can use it in the formula for the area of the isosceles triangle. Let's note: Height FG=5FG=5
Now let's see that we have only half of the base CG=4CG =4 .
Since FGFG is given as the median, we can deduce that also GB=4GB=4 and consequently, the entire side of the base CD=8CD=8
Now let's put it in the formula:
4×82=16\frac{4\times 8}{2}=16
The area of the triangle FDCFDC is 1616 cm2 .


Do you know what the answer is?

Bonus exercise (tip) for advanced level

Formula to calculate the area of an isosceles triangle that is also a right triangle:

If you come across calculating the area of an isosceles triangle whose height has not been given, but you know it is a right triangle, it is useful to know the following trick:

A5 - Area of an isosceles triangle that is also a right triangle

Let's see how it is done by applying it in an exercise: Before you, you have an isosceles right triangle ABCABC
Given that AB=ACAB=AC
angle ABC=90ABC = 90
AB=4AB=4

Calculate the area of the triangle

Solution: Let's not be scared of not having data about the height and proceed according to the formula: the triangle is isosceles, therefore AB=AC=3AB=AC=3.

These are the two legs of the triangle - they form a right angle. Consequently, we will obtain:
4×42=8 \frac{4\times4}{2}=8
The area of the triangle is 88 cm2 .


Examples and exercises with solutions for the area of an isosceles triangle

Exercise #1

Calculate the area of the right triangle below:

101010666888AAACCCBBB

Video Solution

Step-by-Step Solution

Due to the fact that AB is perpendicular to BC and forms a 90-degree angle,

it can be argued that AB is the height of the triangle.

Hence we can calculate the area as follows:

AB×BC2=8×62=482=24 \frac{AB\times BC}{2}=\frac{8\times6}{2}=\frac{48}{2}=24

Answer

24 cm²

Exercise #2

Calculate the area of the following triangle:

4.54.54.5777AAABBBCCCEEE

Video Solution

Step-by-Step Solution

To find the area of the triangle, we will use the formula for the area of a triangle:

Area=12×base×height \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

From the problem:

  • The length of the base BC BC is given as 7 units.
  • The height from point A A perpendicular to the base BC BC is given as 4.5 units.

Substitute the given values into the area formula:

Area=12×7×4.5 \text{Area} = \frac{1}{2} \times 7 \times 4.5

Calculate the expression step-by-step:

Area=12×31.5 \text{Area} = \frac{1}{2} \times 31.5

Area=15.75 \text{Area} = 15.75

Therefore, the area of the triangle is 15.75 15.75 square units. This corresponds to the given choice: 15.75 15.75 .

Answer

15.75

Exercise #3

What is the area of the triangle in the drawing?

5557778.68.68.6

Video Solution

Step-by-Step Solution

First, we will identify the data points we need to be able to find the area of the triangle.

the formula for the area of the triangle: height*opposite side / 2

Since it is a right triangle, we know that the straight sides are actually also the heights between each other, that is, the side that measures 5 and the side that measures 7.

We multiply the legs and divide by 2

5×72=352=17.5 \frac{5\times7}{2}=\frac{35}{2}=17.5

Answer

17.5

Exercise #4

Calculate the area of the triangle using the data in the figure below.

666888AAABBBCCC

Video Solution

Step-by-Step Solution

To calculate the area of the triangle, we will follow these steps:

  • Identify the base, CB, as 6 units.
  • Identify the height, AC, as 8 units.
  • Apply the area formula for a triangle.

Now, let's work through these steps:

The triangle is a right triangle with base CB=6 CB = 6 units and height AC=8 AC = 8 units.

The area of a triangle is determined using the formula:

Area=12×base×height \text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

Substituting the known values, we have:

Area=12×6×8 \text{Area} = \frac{1}{2} \times 6 \times 8

Perform the multiplication and division:

Area=12×48=24 \text{Area} = \frac{1}{2} \times 48 = 24

Therefore, the area of the triangle is 24 24 square units.

Answer

24

Exercise #5

Calculate the area of the triangle below, if possible.

7.67.67.6444

Video Solution

Step-by-Step Solution

To solve this problem, we begin by analyzing the given triangle in the diagram:

While the triangle graphic suggests some line segments labeled with the values "7.6" and "4", it does not confirm these as directly usable as pure base or height without additional proven inter-contextual relationships establishing perpendicularity or side/unit equivalences.

Without a clear base and perpendicular height value, we cannot apply the triangle's area formula Area=12×base×height \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} effectively, nor do we have all side lengths for Heron's formula.

Therefore, due to insufficient information that specifically identifies necessary dimensions for area calculations such as clear height to a base or all sides' measures, the area of this triangle cannot be calculated.

The correct answer to the problem, based on insufficient explicit calculable details, is: It cannot be calculated.

Answer

It cannot be calculated.

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