Triangle Area Calculation: Find Area with Perimeter 26 and Height 6

Q: Why can't I just use the perimeter to find the area directly?

Perimeter and area measure different things! Perimeter is the distance around the triangle, while area measures the space inside. You need the base length (one specific side) for the area formula, not the total perimeter.

Q: How do I know which side to use as the base?

Use the side that the height is perpendicular to. In this problem, the height of 6 is drawn perpendicular to side BC, so BC is your base. The diagram shows this clearly with the vertical line.

Q: What if I calculated the wrong side length?

Always check: side 1 + side 2 + side 3 = perimeter. Here: 9 + 7 + 10 = 26 ✓. If your three sides don't add up to the given perimeter, recalculate the missing side.

Q: Can I use a different side as the base?

You could, but you'd need the height perpendicular to that side. Since the problem gives you the height to BC (which is 6), it's easiest to use BC = 10 as your base.

Q: Why do we divide by 2 in the area formula?

The triangle area formula $ \frac{base \times height}{2} $ comes from the fact that a triangle is half of a rectangle. If you made a rectangle with the same base and height, the triangle would be exactly half its area.

Triangle Area with Perimeter-Height Method

Calculate the area of the

triangle ABC given that its

perimeter equals 26.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of the triangle ABC

00:03 The perimeter of the triangle equals the sum of its sides

00:12 Substitute in the relevant values and calculate to find BC

00:23 Isolate BC

00:29 Now we have the length of the base BC

00:38 Apply the formula for calculating the area of a triangle

00:41 (base(BC) x height(AE)) divided by 2

00:48 Substitute in the relevant values and proceed to solve

01:01 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the area of the

triangle ABC given that its

perimeter equals 26.

Step-by-step solution

Remember that the perimeter of a triangle is equal to the sum of all of the sides together,

We begin by finding side BC:

$26=9+7+BC$

$26=16+BC$

We then move the 16 to the left section and keep the corresponding sign:

$26-16=BC$

$10=BC$

We use the formula to calculate the area of a triangle:

(the side * the height) /2

That is:

$\frac{BC\times AE}{2}$

Lastly we insert the existing data:

$\frac{10\times6}{2}=\frac{60}{2}=30$

Final Answer

Key Points to Remember

Essential concepts to master this topic

Perimeter Rule: Sum of all three sides equals given perimeter
Technique: Find missing side: 26 - 9 - 7 = 10 units
Check: Verify area formula: $\frac{base \times height}{2} = \frac{10 \times 6}{2} = 30$ ✓

Common Mistakes

Avoid these frequent errors

Using perimeter instead of base in area formula
Don't substitute the perimeter 26 directly into area = base × height ÷ 2 = wrong answer of 78! The perimeter includes all sides, but area needs only the base. Always find the missing side first, then use that base in the area formula.

Practice Quiz

Test your knowledge with interactive questions

Angle A is equal to 30°.
Angle B is equal to 60°.
Angle C is equal to 90°.

Can these angles form a triangle?

FAQ

Everything you need to know about this question

Why can't I just use the perimeter to find the area directly?

Perimeter and area measure different things! Perimeter is the distance around the triangle, while area measures the space inside. You need the base length (one specific side) for the area formula, not the total perimeter.

How do I know which side to use as the base?

Use the side that the height is perpendicular to. In this problem, the height of 6 is drawn perpendicular to side BC, so BC is your base. The diagram shows this clearly with the vertical line.

What if I calculated the wrong side length?

Always check: side 1 + side 2 + side 3 = perimeter. Here: 9 + 7 + 10 = 26 ✓. If your three sides don't add up to the given perimeter, recalculate the missing side.

Can I use a different side as the base?

You could, but you'd need the height perpendicular to that side. Since the problem gives you the height to BC (which is 6), it's easiest to use BC = 10 as your base.

Why do we divide by 2 in the area formula?

The triangle area formula $\frac{base \times height}{2}$ comes from the fact that a triangle is half of a rectangle. If you made a rectangle with the same base and height, the triangle would be exactly half its area.

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