Views of Solids and Cross-Sections
Theory
A view of a solid is the flat shape you see when you look straight at one side of it. A cross-section is the flat face you get when you cut straight through it. This Year 7 maths page shows front, side and top views of solids made of cubes, counting hidden cubes, and the cross-sections of prisms, pyramids, cylinders and cones.
Front, side and back views. Look straight at one side. Each column shows its tallest stack, because the stacks behind it are hidden. From the right side, the front of the solid is on your left. The back view is the front view flipped left to right.
Top view (plan view). Look straight down. Each stack shows as one square, however tall. A plan can show the height of each stack as a number.
Counting cubes. Add the heights of all the stacks. Every cube sits on the table or on another cube, so some cubes are hidden underneath or behind.
Views of a prism. Look straight at an end and you see the cross-section. Look at a side or from above and you see rectangles, one for each face you can see.
Cross-sections. A prism has a uniform cross-section: every cut parallel to its end face (its base, even when the prism lies on its side) is the same shape and size as that end. A prism is named after its cross-section, such as a triangular prism or a pentagonal prism. Pyramids and cones are not prisms: their cuts get smaller towards the apex.
Cross-sections and views:
| Solid | Cut | Cross-section |
|---|---|---|
| cylinder | parallel to the circular end | circle |
| cone | straight down through the apex | triangle |
| square pyramid | parallel to the base | square (smaller) |
| cube | parallel to a face / slanted / corner sliced off | square / rectangle / triangle |
| Solid | Front view | Top view |
|---|---|---|
| cylinder standing up | rectangle | circle |
| cylinder on its side, end at the front | circle | rectangle |
| triangular prism, end at the front | triangle | rectangle |
| cone standing up | triangle | circle |
How to draw the views of a solid made of cubes
- Front: for each column (left to right), draw the tallest stack.
- Right: for each row (front row on the left), draw the tallest stack.
- Top: draw one square for every stack, front at the bottom.
How to count the cubes
- Find the height of every stack (use a plan if you have one).
- Add the heights, including hidden cubes underneath.
The front view is the end, a pentagon, so it is a pentagonal prism. Top: a rectangle with a line along the middle (the ridge). Right: a rectangle with a line across it (the edge of the roof).
Front: a circle. Top: a rectangle. Right: a rectangle.
| \(2 + 4 + 2\) | \(=\) | \(8\) |
| \(1 + 2 + 1\) | \(=\) | \(4\) |
| \(8 + 4\) | \(=\) | \(12\) |
\(12\) cubes. Front view columns: \(2, 4, 2\). Right view (front row on the left): \(2, 4\).
Front: the taller stack in each column, \(2, 2, 3\). Right: the tallest in each row, front row first, \(2, 3\). Top: \(6\) squares (\(3\) by \(2\)).
| \((2 + 1) + (3 + 1) + 1\) | \(=\) | \(8\) |
| \((2 + 2) + (3 + 3) + 1\) | \(=\) | \(11\) |
Least: in each column one stack is tall and the other is \(1\) cube: \(8\). Greatest: both stacks are tall: \(11\).
| \(\dfrac{1}{2} \times 8 \times 5\) | \(=\) | \(20\) |
The cross-section is uniform, so the slice is the same triangle: \(20~\text{cm}^2\). (The \(12\) cm length is not needed.)
Common pitfalls
Frequently asked questions
What is a plan view?
The top view: what you see looking straight down.
What is a cross-section?
The flat face made by cutting straight through a solid.
What is a uniform cross-section?
Every cut parallel to the end is the same shape and size, as in a prism.
How is a prism named?
By the shape of its cross-section, such as a triangular prism.
Can a cube have a triangular cross-section?
Yes, if a corner is sliced off.