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Math  /  Geometry

QuestionWhat is the volume of this triangular pyramid?

Studdy Solution

STEP 1

What is this asking? We need to find the volume of a pyramid with a triangular base! Watch out! Don't mix up the height of the triangle at the base with the height of the *whole* pyramid.
Also, don't forget that a triangle's area is *half* of base times height.

STEP 2

1. Find the area of the triangular base.
2. Calculate the pyramid's volume.

STEP 3

Alright, let's **start** by finding the area of the triangle at the bottom of our pyramid.
Remember, the area of a triangle is 12⋅base⋅height\frac{1}{2} \cdot \text{base} \cdot \text{height}.

STEP 4

We know the **base** of the triangle is 9393 mm and its **height** is 5050 mm.
Plugging those values into our formula, we get: Area=12⋅93 mm⋅50 mm \text{Area} = \frac{1}{2} \cdot 93 \text{ mm} \cdot 50 \text{ mm}

STEP 5

Let's crunch those numbers!
Half of 9393 is 46.546.5, and 46.546.5 times 5050 is 23252325.
So, the **area of the triangular base** is 2325 mm22325 \text{ mm}^2.
Don't forget those units--millimeters squared!

STEP 6

Now for the main event: finding the **volume of the pyramid**!
The formula for the volume of a pyramid is 13⋅base area⋅height\frac{1}{3} \cdot \text{base area} \cdot \text{height}.
Notice that "base area" means the area of the triangle we *just* calculated.

STEP 7

We already know the **base area** is 2325 mm22325 \text{ mm}^2, and the problem tells us the **pyramid's height** is 9898 mm.
Let's plug those values into our volume formula: Volume=13⋅2325 mm2⋅98 mm \text{Volume} = \frac{1}{3} \cdot 2325 \text{ mm}^2 \cdot 98 \text{ mm}

STEP 8

Time for some more exciting calculations!
One third of 23252325 is 775775.
Now, 775775 multiplied by 9898 gives us 7595075950.
Therefore, the **volume of the triangular pyramid** is 75950 mm375950 \text{ mm}^3.
Remember, it's millimeters *cubed* this time, since we're talking about volume!

STEP 9

The volume of the triangular pyramid is 75950 mm375950 \text{ mm}^3.

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