Depth · one concept from first principles · step 2 of 4
Volume is the base stacked to the height
What is the volume of the shaded solid inside the rectangular prism at the right, in ?
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — A prism's volume is always its base area times its height, whatever shape the base is. So the only real work is reading the triangle's base and height off the box -- and the box's depth is the triangle's height.
1STEP 1
Read the triangle off the box
The triangle's base lies along the box's 17 cm edge, and its apex reaches the opposite wall, so its height is the box's depth of 10 cm.
Base 17 cm, height 10 cm.
6.G.A.1Visualize Spatial Relationships2STEP 2
Find the base area
A triangle is half of the rectangle on the same base and height.
17 × 10 ÷ 2 = 85
The base covers 85 cm².
6.G.A.1Draw A Diagram3STEP 3
Multiply by the height
The shaded solid rises the full 24 cm of the box, and a prism's volume is base area times height.
85 × 24 = 2040
The shaded solid is 2040 cm³.
6.G.A.2Identify SubproblemsA prism's volume is its base area times its height, whatever the base's shape.
Why?
A prism is the same base shape carried straight up, so it stacks into layers that are all copies of the base.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Each layer holds as much as the base's area, and the height says how many layers there are.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
Answer
2040 cm³
Compare with the box: it holds 17 × 10 × 24 = 4080 cm³, and the shaded solid is exactly half of that, just as its triangular base is half the box's floor.
Takeaway
A prism's volume is its base area times its height, whatever shape the base is.
- Read the triangle off the box
- Find the base area
- Multiply by the height
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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