The base of a triangular pyramid is a triangle with corners at #(6 ,5 )#, #(7 ,1 )#, and #(4 ,7 )#. If the pyramid has a height of #7 #, what is the pyramid's volume?

Question

The base of a triangular pyramid is a triangle with corners at #(6  ,5  )#, #(7  ,1   )#, and #(4  ,7  )#.  If the pyramid has a height of #7  #, what is the pyramid's volume?

Wyatt Anderson · Accepted Answer

Volume of the pyramid is #6.95# cubic unit

The sides of triangular base #A=sqrt((6-7)^2+(5-1)^2)=sqrt17=4.12#; #B=sqrt((7-4)^2+(1-7)^2)=sqrt45=6.71#; #C=sqrt((4-6)^2+(7-5)^2)=sqrt8=2.83#; Semi perimeer #s=(4.12+6.71+2.83)/2=6.83# Area of triangular base#A_t=sqrt(6.83*(6.83-4.12)(6.83-6.71)(6.83-2.83))=2.98# Volume of the pyramid is #V=(1/3)*A_t*ht = (2.98*7)/3=6.95# cubic unit[Ans}

Carson Arnold · Answer

undefined To find the volume of a triangular pyramid, you can use the formula:

Volume = (1/3) * Area of Base * Height

First, calculate the area of the base triangle using the coordinates given for its vertices. You can use the formula for the area of a triangle given its vertices:

Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Once you have the area of the base triangle, multiply it by the height of the pyramid and divide the result by 3 to find the volume.

HIX Tutor · Answer

undefined To find the volume of the triangular pyramid, follow these steps:

1. **Find the area of the triangular base**:
   - Use the formula:  
     $$ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} $$
   - First, find the lengths of the sides. The vertices are:
     - Point A (6, 5)
     - Point B (7, 1)
     - Point C (4, 7)
   - Use the formula for area from the coordinates:
     $$
     \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
     $$
     where:
     - $ (x_1, y_1) = (6, 5) $
     - $ (x_2, y_2) = (7, 1) $
     - $ (x_3, y_3) = (4, 7) $

- Plug in the values:
     $$
     \text{Area} = \frac{1}{2} \left| 6(1 - 7) + 7(7 - 5) + 4(5 - 1) \right|
     $$
     $$
     = \frac{1}{2} \left| 6 \times (-6) + 7 \times 2 + 4 \times 4 \right|
     $$
     $$
     = \frac{1}{2} \left| -36 + 14 + 16 \right|
     $$
     $$
     = \frac{1}{2} \left| -36 + 30 \right|
     $$
     $$
     = \frac{1}{2} \left| -6 \right| = \frac{1}{2} \times 6 = 3
     $$

2. **Use the area to find the volume**:
   - The formula for the volume of a pyramid is:
     $$
     \text{Volume} = \frac{1}{3} \times \text{Base Area} \times \text{Height}
     $$
   - The height is 7.
   - Plug in the area and height:
     $$
     \text{Volume} = \frac{1}{3} \times 3 \times 7
     $$
     $$
     = \frac{21}{3} = 7
     $$

3. **Final answer**:
   - The volume of the pyramid is **7 cubic units**.