What is the area of a triangle with sides of length 12 cm, 13 cm and 5 cm?
30 cm 2
Finding the Area of a Triangle with Sides 12 cm, 13 cm, and 5 cm
The problem asks us to find the area of a triangle given its side lengths: 12 cm, 13 cm, and 5 cm. When we are given the side lengths of a triangle, the first step is often to check if it is a special type of triangle, such as a right-angled triangle. This can make calculating the area much simpler.
Checking for a Right-Angled Triangle using the Pythagorean Theorem
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides (legs). The formula is expressed as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
In this triangle, the side lengths are 5 cm, 12 cm, and 13 cm. The longest side is 13 cm, so if it is a right triangle, 13 cm would be the hypotenuse. Let's check if the Pythagorean theorem holds true for these side lengths:
- Let \(a = 5\) cm
- Let \(b = 12\) cm
- Let \(c = 13\) cm
Now, we substitute these values into the Pythagorean theorem formula:
\(a^2 + b^2 = 5^2 + 12^2\)
\(5^2 = 25\)
\(12^2 = 144\)
So, \(a^2 + b^2 = 25 + 144 = 169\)
Next, we calculate the square of the longest side, \(c\):
\(c^2 = 13^2 = 169\)
Since \(a^2 + b^2 = 169\) and \(c^2 = 169\), we have \(a^2 + b^2 = c^2\).
This confirms that the triangle with sides 5 cm, 12 cm, and 13 cm is indeed a right-angled triangle. The sides of length 5 cm and 12 cm are the legs, which can serve as the base and height of the triangle.
Calculating the Area of the Right-Angled Triangle
The area of a right-angled triangle is given by the formula:
Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)
Using the legs as the base and height:
- Base = 12 cm
- Height = 5 cm
Now, we calculate the area:
Area = \(\frac{1}{2} \times 12 \text{ cm} \times 5 \text{ cm}\)
Area = \(\frac{1}{2} \times 60 \text{ cm}^2\)
Area = \(30 \text{ cm}^2\)
Therefore, the area of the triangle with sides 12 cm, 13 cm, and 5 cm is 30 cm\(^2\).
Revision Table: Triangle Area Calculation
| Concept | Description | Formula/Method Used |
|---|---|---|
| Triangle Type | Determining if the triangle is right-angled. | Pythagorean Theorem (\(a^2 + b^2 = c^2\)) |
| Base and Height | Identifying the perpendicular sides in a right triangle. | The two shorter sides (legs). |
| Area Calculation | Using the base and height to find the area. | Area = \(\frac{1}{2} \times \text{base} \times \text{height}\) |
Additional Information: Heron's Formula for Triangle Area
If the triangle were not a right-angled triangle, or if you weren't sure, you could use Heron's formula to find the area when all three side lengths are known. Heron's formula works for any triangle.
Heron's Formula involves two steps:
- Calculate the semi-perimeter (\(s\)) of the triangle. The semi-perimeter is half the perimeter: \(s = \frac{a + b + c}{2}\)
- Calculate the area using the formula: Area = \(\sqrt{s(s-a)(s-b)(s-c)}\)
Let's apply Heron's formula to our triangle with sides 5, 12, and 13 cm to confirm the result:
- \(a = 5\), \(b = 12\), \(c = 13\)
- Semi-perimeter \(s = \frac{5 + 12 + 13}{2} = \frac{30}{2} = 15\) cm
Now, calculate the terms inside the square root:
- \(s-a = 15 - 5 = 10\)
- \(s-b = 15 - 12 = 3\)
- \(s-c = 15 - 13 = 2\)
Finally, apply Heron's formula:
Area = \(\sqrt{15 \times 10 \times 3 \times 2}\)
Area = \(\sqrt{15 \times 60}\)
Area = \(\sqrt{900}\)
Area = \(30 \text{ cm}^2\)
As expected, Heron's formula gives the same area, confirming our result obtained by identifying it as a right triangle.
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