Area of Equilateral Triangles - Examples, Exercises and Solutions

This question is a bit confusing. We need start by identifying which parts of the data are relevant to us.

Remember the formula for the area of a triangle:

The height is a straight line that comes out of an angle and forms a right angle with the opposite side.

In the drawing we have a height of 6.

It goes down to the opposite side whose length is 5.

And therefore, these are the data points that we will use.

We replace in the formula:

$\frac{6\times5}{2}=\frac{30}{2}=15$

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

$\frac{5\times7}{2}=\frac{35}{2}=17.5$

The triangle we are looking at is the large triangle - ABC

The triangle is formed by three sides AB, BC, and CA.

Now let's remember what we need for the calculation of a triangular area:

(side x the height that descends from the side)/2

Therefore, the first thing we must find is a suitable height and side.

We are given the side AC, but there is no descending height, so it is not useful to us.

The side AB is not given,

And so we are left with the side BC, which is given.

From the side BC descends the height AD (the two form a 90-degree angle).

It can be argued that BC is also a height, but if we delve deeper it seems that CD can be a height in the triangle ADC,

and BD is a height in the triangle ADB (both are the sides of a right triangle, therefore they are the height and the side).

As we do not know if the triangle is isosceles or not, it is also not possible to know if CD=DB, or what their ratio is, and this theory fails.

Let's remember again the formula for triangular area and replace the data we have in the formula:

(side* the height that descends from the side)/2

Now we replace the existing data in this formula:

$\frac{CB\times AD}{2}$

$\frac{11.6\times3}{2}$

$\frac{34.8}{2}=17.4$

To solve for the area of a triangle when the base and height are given, we'll use the formula:

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

Given:

• Base = $4$ units

• Height = $7$ units

Apply the formula:

$\begin{aligned} \text{Area} &= \frac{1}{2} \times 4 \times 7 \\ &= \frac{1}{2} \times 28 \\ &= 14 \end{aligned}$

Thus, the area of the triangle is $14$ square units.

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

• Step 1: Identify the given base and height from the problem.
• Step 2: Apply the formula for the area of a triangle.
• Step 3: Calculate the area by substituting the values into the formula.

Let's work through the problem:

Step 1: The base $|AB|$ of the triangle is given as 8 units, and the height $|BC|$ is 6 units.

Step 2: The formula for the area of a triangle is:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

Step 3: Substitute the given values into the formula:

$A = \frac{1}{2} \times 8 \times 6$

Perform the multiplication:

$A = \frac{1}{2} \times 48 = 24$

Therefore, the area of the triangle is $\mathbf{24}$ square units.

To solve this problem, we'll follow these steps:

• Step 1: Identify the given information
• Step 2: Apply the appropriate formula
• Step 3: Perform the necessary calculations

Now, let's work through each step:

Step 1: We are given that $AC = 9$ (the height) and $BC = 10$ (the base) of the triangle.

Step 2: We'll use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

Step 3: Plugging in our values, we have:

Therefore, the area of the triangle is $45$.

To solve the problem of finding the area of triangle $\triangle ABC$, we follow these steps:

• Step 1: Identify the given measurements.
• Step 2: Use the appropriate formula for the area of a triangle.
• Step 3: Calculate the area using these measurements.

Let's go through each step in detail:
Step 1: From the figure, the base $AB = 10$ and height $AC = 2$.
Step 2: The formula for the area of a triangle is: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
Step 3: Substituting the known values into the formula, we get:

$\text{Area} = \frac{1}{2} \times 10 \times 2 = \frac{1}{2} \times 20 = 10$

Therefore, the area of triangle $\triangle ABC$ is 10.

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 $CB = 6$ units and height $AC = 8$ units.

The area of a triangle is determined using the formula:

Substituting the known values, we have:

Perform the multiplication and division:

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

To solve this problem, begin by identifying the elements involved in calculating the area of a right triangle. In a right triangle, the two sides that form the right angle are known as the legs. These legs act as the base and height of the triangle.

The formula for the area of a triangle is given by:

In the case of a right triangle, the base and height are the two legs. Therefore, the process of finding the area involves multiplying the lengths of the two legs together and then dividing the product by 2.

Based on this analysis, the correct way to complete the sentence in the problem is:

To find the area of a right triangle, one must multiply the two legs by each other and divide by 2.

To solve this problem, we'll follow these steps:

• Step 1: Identify the base and height of the triangle.
• Step 2: Substitute these values into the formula for the area of a triangle.
• Step 3: Perform the calculation to determine the area.

Let's work through each step:

Step 1: From the given information, the base of the triangle is 7 units, and the height is 4 units.

Step 2: We'll use the formula for the area of a triangle:
$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Step 3: Plugging in the values for the base and height, we have:
$\text{Area} = \frac{1}{2} \times 7 \times 4$

Performing the multiplication, we get:
$\text{Area} = \frac{1}{2} \times 28 = 14$

Therefore, the area of the triangle is $\textbf{14}$ square units.

To solve this problem, we will determine the area of the triangle using the given base and height. Here are the steps:

• Identify the given base and height: base $= 6$, height $= 3.5$.
• Apply the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
• Substitute the given values into the formula: $\text{Area} = \frac{1}{2} \times 6 \times 3.5$.
• Calculate the area: $= \frac{1}{2} \times 21 = 10.5$.

Therefore, the area of the triangle is $10.5$, which matches the correct multiple-choice option provided.

The formula to calculate the area of a triangle is:

(side * height corresponding to the side) / 2

Note that in the triangle provided to us, we have the length of the side but not the height.

That is, we do not have enough data to perform the calculation.

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

To solve this problem, we'll follow these steps:

• Step 1: Identify the given base and height of the triangle.
• Step 2: Apply the formula for the area of a triangle.
• Step 3: Perform the necessary calculations to find the area.

Now, let's work through each step:
Step 1: The base of the triangle is given as 7 units, and the height is given as 5 units.
Step 2: We'll use the formula for the area of a triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.
Step 3: Plugging in our values, we have:
$\text{Area} = \frac{1}{2} \times 7 \times 5 = \frac{1}{2} \times 35 = 17.5$.

Therefore, the area of the triangle is $17.5$ square units.

To solve this problem, we will follow these steps:

• Step 1: Identify the relevant sides based on problem context
• Step 2: Apply the standard triangle area formula
• Step 3: Calculate the area based on the known values for base and height

Let's work through each step in detail:

Step 1: We are seeking to calculate the area of the triangle. We identified that the line segment of 4 units represents the height, and the base is 7 units.

Step 2: We will apply the formula for the area of a right triangle: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$.

Step 3: Plug the values we have: $\text{Area} = \frac{1}{2} \times 7 \times 4 = \frac{1}{2} \times 28 = 14$.

Thus, the area of the triangle is $14$ square units.