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The **area** of a triangle measures the amount of surface it encloses. Computing it requires knowing a **base** ($$b$$) and the corresponding **height** ($$h$$). The height is a perpendicular segment drawn from the base to the opposite vertex, forming a $$90°$$ angle with the base. The formula is:
$$A = \frac{1}{2}bh$$
In words, the area of a triangle equals one-half the product of its base and its height.

Area of a Triangle

A graphic designer is calculating the total surface area of a triangular element for a company logo. After multiplying the length of the base by the perpendicular height, what final mathematical operation must the designer perform to find the correct area ($A$) according to the geometric formula?

Calculating Area for Graphic Design

Based on the scenario provided, what is the formal geometric definition of the 'height' ($$h$$) that the sailmaker must identify to correctly calculate the area of the jib sail?

Material Estimation for Custom Sail Design

Apply the geometry problem-solving strategy to find the height of a triangle when its area and base are known.

**Problem:** The area of a triangular painting is $$126$$ square inches. The base is $$18$$ inches. What is the height?

1. **Read:** Determine that the problem involves a triangle with an area of $$126$$ square inches and a base of $$18$$ inches. Draw and label the figure.
2. **Identify:** The height of the triangle.
3. **Name:** Let $$h =$$ the height. (See the accompanying figure for the labeled area and base.)
4. **Translate:** Write the triangle area formula and substitute the known values:
$$A = \frac{1}{2}bh$$
$$126 = \frac{1}{2} \cdot 18 \cdot h$$
5. **Solve:** Simplify the right side: $$126 = 9h$$. Divide both sides by $$9$$: $$14 = h$$.
6. **Check:** Verify that an area of $$126$$ results from a base of $$18$$ and height of $$14$$: $$\frac{1}{2} \cdot 18 \cdot 14 = 9 \cdot 14 = 126$$ $$\checkmark$$
7. **Answer:** The height of the triangle is $$14$$ inches.

Finding the Height of a Triangle Given Its Area and Base

Apply the geometry problem-solving strategy to find the base of a triangle when its area and height are known.

**Problem:** A triangular tent door has an area of $$15$$ square feet. The height is $$5$$ feet. What is the length of the base?

1. **Read:** A triangle has an area of $$15$$ sq. ft. and a height of $$5$$ ft. Draw and label the figure.
2. **Identify:** The length of the base.
3. **Name:** Let $$b =$$ the length of the base.
4. **Translate:** Write the triangle area formula and substitute the given values:
$$A = \frac{1}{2}bh$$
$$15 = \frac{1}{2} \cdot b \cdot 5$$
5. **Solve:** Multiply both sides by $$2$$ to clear the fraction: $$30 = 5b$$. Divide by $$5$$ to completely isolate $$b$$: $$b = 6$$.
6. **Check:** Using the area formula, verify that $$\frac{1}{2} \cdot 6 \cdot 5 = 3 \cdot 5 = 15$$ $$\checkmark$$
7. **Answer:** The length of the base is $$6$$ feet.

Finding the Base of a Triangle Given Its Area and Height

What does the area of a triangle measure?

What angle does a triangle's height form with its base?

The area of a triangle is $$A=bh$$.

Match each part of $$A=\frac{1}{2}bh$$ with its meaning.

Order the steps for finding the area of a triangle.

A triangle's height is _____ to its base.

Which formula gives the area of a triangle with base $$b$$ and height $$h$$?

Question:
State the triangle area formula. How must the height relate to the base?

Sample answer:
$$A=\frac{1}{2}bh$$. The height must be perpendicular to the base, forming a $$90°$$ angle.

Key points:
- $$A=\frac{1}{2}bh$$
- $$b$$ is the base.
- $$h$$ is the height.
- The height is perpendicular to the base.

Rubric:
Full credit: gives the correct formula and states that the height is perpendicular to the base. Partial credit: gives only one correct part.

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