Triangle Area Calculation: Given Perimeter 24 and Height 5

Q: How do I know which measurement is the height?

The height is always the perpendicular distance from a vertex to the opposite side. In diagrams, look for the dotted line or right angle symbol showing the height.

Q: What if the perimeter and height don't match my calculation?

Double-check that you're using side lengths for perimeter (not height) and height for area (not side length). These are different measurements!

Q: Why is my area answer a decimal?

Triangle areas can be decimals! When you multiply by $ \frac{1}{2} $, you often get decimal results. 22.5 is a perfectly valid area measurement.

Q: Do I always need all three sides to find area?

No! You only need the base and height for area. But if you're given perimeter, you can find missing sides using: missing side = perimeter - sum of known sides.

Triangle Area with Perimeter Constraint

Calculate the area of the triangle ABC:

Given that: Perimeter=24

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the area of triangle ABC

00:02 The perimeter of the triangle equals the sum of its sides

00:10 Substitute in the relevant values and proceed to calculate to determine BC

00:22 Isolate BC

00:31 Now we have the length of base BC

00:37 Apply the formula for calculating the triangle's area

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

00:50 Substitute in the relevant values and proceed to solve

01:02 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: Perimeter=24

Step-by-step solution

To find the area of triangle $\triangle ABC$ , follow these steps:

Step 1: Since the perimeter is 24, and using the side lengths $AB = 8$ , $AC = 7$ , we calculate the third side $BC$ . Using $AB + AC + BC = 24$ , we find:

8 + 7 + BC = 24 \implies BC = 24 - 15 = 9

Step 2: Given that the height from point $A$ to base $BC$ is 5, use the triangle area formula:

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

Substituting base $BC = 9$ and height = 5:

\text{Area} = \frac{1}{2} \times 9 \times 5 = \frac{45}{2} = 22.5

Therefore, the area of triangle $\triangle ABC$ is $22.5$ .

Final Answer

22.5

Key Points to Remember

Essential concepts to master this topic

Perimeter Rule: Sum of all three sides equals given perimeter
Technique: Use $BC = 24 - 8 - 7 = 9$ to find missing side
Check: Verify $\frac{1}{2} \times 9 \times 5 = 22.5$ using area formula ✓

Common Mistakes

Avoid these frequent errors

Using height as a side length
Don't substitute the height value (5) as one of the triangle's sides = wrong perimeter calculation! Height is perpendicular distance, not a side. Always distinguish between side lengths and perpendicular height when calculating area.

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

How do I know which measurement is the height?

The height is always the perpendicular distance from a vertex to the opposite side. In diagrams, look for the dotted line or right angle symbol showing the height.

Can I use any side as the base?

Yes! You can choose any side as the base, but you must use the corresponding height - the perpendicular distance from the opposite vertex to that base.

What if the perimeter and height don't match my calculation?

Double-check that you're using side lengths for perimeter (not height) and height for area (not side length). These are different measurements!

Why is my area answer a decimal?

Triangle areas can be decimals! When you multiply by $\frac{1}{2}$ , you often get decimal results. 22.5 is a perfectly valid area measurement.

Do I always need all three sides to find area?

No! You only need the base and height for area. But if you're given perimeter, you can find missing sides using: missing side = perimeter - sum of known sides.

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