Question

Find the area.
A = [?]cm²
12 cm, 10 cm and 15 cm
Shape: Triangle
Area of a triangle: A = bh/2

Answer 1

To calculate the area of a triangle using the formula \( A = \frac{bh}{2} \), we typically need the base \( b \) and the height \( h \). However, in your case, it sounds like you are dealing with a triangle where you have the lengths of all three sides: 12 cm, 10 cm, and 15 cm.

When you don't have the height, you can use Heron's formula to calculate the area of the triangle. Heron's formula is given by:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]

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

1. First, calculate the semi-perimeter \( s \):

\[
s = \frac{a + b + c}{2} = \frac{12 + 10 + 15}{2} = \frac{37}{2} = 18.5 \text{ cm}
\]

2. Now we apply the values into Heron's formula:

\[
A = \sqrt{s(s-a)(s-b)(s-c)}
\]
\[
A = \sqrt{18.5(18.5 - 12)(18.5 - 10)(18.5 - 15)}
\]
\[
A = \sqrt{18.5(6.5)(8.5)(3.5)}
\]

3. Now, calculate the values step-by-step:

\[
18.5 - 12 = 6.5
\]
\[
18.5 - 10 = 8.5
\]
\[
18.5 - 15 = 3.5
\]

4. Multiply the terms inside the square root:

\[
A = \sqrt{18.5 \times 6.5 \times 8.5 \times 3.5}
\]

Calculating this step-by-step:

- First multiply \( 6.5 \times 8.5 = 55.25 \)
- Then multiply \( 55.25 \times 3.5 = 193.375 \)
- Finally multiply \( 193.375 \times 18.5 = 3578.4375 \)

Now take the square root:

\[
A = \sqrt{3578.4375} \approx 59.8 \text{ cm}^2
\]

Thus, the area of the triangle is approximately:

\[
A \approx 59.8 \, \text{cm}^2
\]