GCSE Maths: Plans and elevations (Higher)
Part of the GCSE Maths volume, surface area and 3d shapes chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Plans and elevations
Imagine walking round a model made of building blocks. From the front it looks one way, from the side another way, and from above another. Architects and engineers draw exactly these views to describe a solid on flat paper.
The view from above is the plan. The view from the front is the front elevation. The view from the side (we use the right-hand side) is the side elevation. Each view is drawn straight on, so you see flat shapes, never a 3D picture. For a solid made of cubes we draw each view on squared paper, one square for each cube face you can see.
How to find an elevation of a stack of cubes. For the front elevation, look at each column from the left to the right and find the tallest stack: a tall stack hides any shorter stack behind it. For the side elevation, look at each row from the front to the back and find its tallest stack. For the plan, just mark every stack as one square (you can write the number of cubes in it).
Other solids. The views of cylinders, cones, spheres, pyramids and prisms are shapes you already know. A cylinder standing on its circular face looks like a rectangle from the front and the side, and like a circle from above. A cone looks like a triangle from the front and a circle (with a dot at the centre) from above.
Key ideas
- Plan: the view from above. Front elevation: the view from the front. Side elevation: the view from the side.
- Draw only what you can see, straight on, with the right sizes on squared paper.
- For cubes: use the tallest stack in each column (front) or row (side). In the side elevation the front is on the left.
- Cylinder: rectangle, rectangle, circle. Cone: triangle, triangle, circle. Sphere: circle, circle, circle. Square-based pyramid: triangle, triangle, square with diagonals.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 1, 3 → tallest = 3
Column 2 (from the left): stacks 2, 1 → tallest = 2 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 3, 1 → tallest = 3
Back row (view position 2): stacks 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Think about what you would see from above and from the front, then draw it to the correct size.
- Plan: look from above. From above you see the circular base, with the point of the cone in the middle.A circle of radius 3 cm with a dot at its centre
- Front elevation: look from the front. From the side-on view the cone looks like a triangle. Its base is the width of the circle and its height is the height of the cone.Base = 2 × 3 = 6 cm, height = 5 cm: an isosceles triangle
- Draw on the grid. Use one square for 1 cm so the drawings are the right size.
Check. The widest part of both drawings is 6 cm, as it should be. ✓
Watch out. From above a cone is a circle, not a triangle.
Method. Draw each view of the triangular prism straight on, using one grid square for 1 cm.
- Front elevation. Looking at the triangular end, we see the triangle: base 4 cm, height 3 cm.Triangle with base 4 and height 3
- Side elevation. From the side we see a rectangular face: as long as the prism (6 cm) and as tall as the triangle (3 cm).Rectangle 6 cm by 3 cm
- Plan. From above we see the two sloping rectangles side by side, each 2 cm wide (half of the base) and 6 cm long. The line between them is the top edge.Rectangle 4 cm by 6 cm with a line down the middle
- Draw on the grid. One square for 1 cm.
Check. The width is 4 in the front elevation and the plan, the height is 3 in both elevations, and the length is 6 in the side elevation and the plan. ✓
Method. Each stack must not be taller than the front column and not taller than the side row it sits in. Work out the tallest each stack could be (largest total) and the fewest cubes that still match both views.
- Set out the grid of stacks. There are 3 columns (from the front elevation) and 2 rows (from the side elevation), so 6 stacks.Columns: 2, 1, 3. Rows (front to back): 3, 2.
- Largest: make every stack as tall as allowed. A stack can be no taller than its column limit and no taller than its row limit, so use the smaller of the two.Front row (limit 3): min(2,3), min(1,3), min(3,3) = 2, 1, 3 = 6 cubes
Back row (limit 2): min(2,2), min(1,2), min(3,2) = 2, 1, 2 = 5 cubes
Largest total = 6 + 5 = 11 - Smallest: use as few cubes as possible. Every column must still reach its height and every row must still reach its height. Let one stack do two jobs at once: a stack that is the tallest in its column and the tallest in its row saves cubes.One arrangement with the fewest cubes (front row, then back row): [0, 0, 3], [2, 1, 0]
Smallest total = 6
Check. The smallest arrangement has front elevation columns [2, 1, 3] and side elevation rows [3, 2], as required. ✓
Watch out. Check both views for every arrangement you try. A total that matches only one view is not enough.
Watch out
For the side elevation, the front of the solid is on the left when you look from the right. Do not add the numbers in a column: use the tallest stack, because the shorter stacks are hidden behind it. A plan shows one square for each stack, whatever its height.Exercise 19C Plans and elevations
Non-calculatorKey words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 1, 3 → tallest = 3
Column 2 (from the left): stacks 2, 1 → tallest = 2 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 3, 1 → tallest = 3
Back row (view position 2): stacks 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand above the solid and look straight down. The plan is the view from above. The back row is at the top of the drawing and the front row is at the bottom.
- Mark every position that has at least one cube. From above you see the top of every stack, however tall it is.Row 1 (from the back): 1, 2
Row 2 (from the back): 3, 1 - Draw the outline of each stack. Each stack is one square. Write the number of cubes in it if you want to show the heights.Total number of cubes = 7
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. Count the cubes in the plan numbers: total 7. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 0, 1 → tallest = 1
Column 2 (from the left): stacks 1, 1 → tallest = 1
Column 3 (from the left): stacks 2, 3 → tallest = 3 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 1, 1, 3
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 1 + 1 + 3 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 1, 1, 3 → tallest = 3
Back row (view position 2): stacks 0, 1, 2 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Add the numbers in all the stacks.
- Read the number in each stack. The back row is 2, 1, 0 and the front row is 1, 3, 1.2, 1, 0 and 1, 3, 1
- Add them. Add the back row, then the front row.(2 + 1 + 0) + (1 + 3 + 1) = 3 + 5 = 8
Check. Add in a different order: 2 + 1 + 1 + 3 + 1 = 8. ✓
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand in front of the solid, looking at it straight on. The front elevation is the view from the front. Each column of the plan lines up from left to right.
- For each column, find the tallest stack. The tallest stack hides any shorter stack behind it, so only the tallest shows.Column 1 (from the left): stacks 2, 1 → tallest = 2
Column 2 (from the left): stacks 1, 3 → tallest = 3
Column 3 (from the left): stacks 0, 1 → tallest = 1 - Draw the elevation on squared paper. In each column, shade as many squares as the tallest stack has cubes.Heights from left to right: 2, 3, 1
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 6 squares: 2 + 3 + 1 = 6. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Key words. A plan is the view of a solid from directly above. An elevation is a view from the side (front elevation: from the front; side elevation: from the side). Pictures of the solid are drawn on a square grid.
Method. An elevation is a view from the side; the plan is the view from above. Only what can be seen is drawn, and a taller stack hides a shorter one behind it.
- Stand to the right of the solid, looking at its right-hand side. The side elevation shows the rows. Looking from the right, the front row is on the left of your view and the back row is on the right.
- For each row, find the tallest stack. The tallest stack in a row hides the shorter ones behind it.Front row (view position 1): stacks 1, 3, 1 → tallest = 3
Back row (view position 2): stacks 2, 1, 0 → tallest = 2 - Draw the elevation on squared paper. Start with the front row on the left. Shade squares up to each row’s tallest stack.Heights from left to right: 3, 2
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The elevation has 5 squares: 3 + 2 = 5. ✓
Watch out. In the side elevation the front row is on the left when you look from the right. Do not add the numbers up: use the tallest in each column or row.
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A tin stands on its circular base.
- Say what you see. From above you look straight down at the circular lid.The view is a circle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A tin stands on its circular base.
- Say what you see. From the front you see the flat side of the tin: as wide as the circle and as tall as the tin.The view is a rectangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A cone has a point at the top and a circle at the bottom.
- Say what you see. From the front the sloping sides meet at the point.The view is a triangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A ball looks the same from every direction.
- Say what you see. From above, the outline of a ball is a circle.The view is a circle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A pyramid comes to a point at the top.
- Say what you see. From the front you see a triangular face (the base is hidden below).The view is a triangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Think about the solid and what you would see from that direction.
- Picture the solid. A cuboid has rectangular faces.
- Say what you see. From above you see the rectangular top face.The view is a rectangle.
Check. Compare with the pictures of the views at the start of this section. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a circle, with a dot where the point is.
- Find the matching picture. The plan of the cone is picture C.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a circle (the lid).
- Find the matching picture. The plan of the cylinder is picture B.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see the square base with lines from each corner to the apex.
- Find the matching picture. The plan of the square-based pyramid is picture A.
Check. The other three plans do not match what you would see from above. ✓
Method. Compare each plan with what you would see looking down on the solid.
- Think about the solid from above. You see a rectangle with the top edge as a line down the middle.
- Find the matching picture. The plan of the triangular prism is picture D.
Check. The other three plans do not match what you would see from above. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The front elevation of a cuboid is a rectangle: as wide as its length and as tall as its height.
- Find the width and height. The 4 cm length faces us and the cuboid is 2 cm high.4 cm wide, 2 cm high
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 8 cm2, the same as a face of 4 by 2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The side elevation is a rectangle: as wide as the cuboid is deep and as tall as its height.
- Find the width and height. From the side we see the 3 cm width, and the height is 2 cm.3 cm wide, 2 cm high
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 6 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The plan is the rectangle you see looking down on the top face.
- Find the sides. The top face is 4 cm long and 3 cm wide.4 cm by 3 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The rectangle has area 12 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. From the side we see a rectangle.
- Find its sides. It is as long as the prism and as tall as the triangle.6 cm by 3 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The area is 18 cm2. ✓
Method. The front elevation is the triangle at the end. Use area = ½ × base × height.
- Substitute. Base 4 cm and height 3 cm.Area = ½ × 4 × 3
- Work it out. 4 × 3 = 12, and half of 12 is 6.Area = 6 cm2
Check. A 4 by 3 rectangle has area 12; the triangle is half of it. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A rectangle means the sides are straight up and down.
- Plan. A circle means it is round from above. A rectangle that is round from above is a cylinder.The solid is a cylinder.
Check. A cuboid would have a rectangle for its plan too. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A triangle means it comes to a point at the top.
- Plan. A circle with a dot means a round base and a point above the centre. That is a cone.The solid is a cone.
Check. A pyramid has a square plan, not a circle. ✓
Method. Match the two views with the solids in the Key ideas box.
- Front elevation. A triangle means it comes to a point at the top.
- Plan. A square with lines from the corners to the middle is the base with the sloping edges going up to the apex.The solid is a square-based pyramid.
Check. A cone has a circle for its plan. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The front elevation of a prism is its cross-section when the end faces you.
- Draw the L-shape. The L is 4 cm wide and 3 cm tall. The bottom arm is 1 cm tall and the upright arm is 1 cm wide.An L made from 4 + 2 = 6 unit squares
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. Area = 4 × 1 + 1 × 2 = 6 cm2. ✓
Key words. A plan is the view of a solid from above. A front elevation is the view from the front and a side elevation is the view from the side. They show only the edges you can see, drawn straight on.
Method. The plan is the view from above: a rectangle with a line where the height changes.
- Find the size. From above the prism is 4 cm wide and 5 cm long.4 cm by 5 cm
- Mark the step. The tall arm is 1 cm wide, so there is a line 1 cm from the left edge.Line at 1 cm
- Here is the finished drawing. Check that your drawing matches it, square for square.
Check. The plan is a rectangle with area 20 cm2. ✓
Method. Each cube has volume 1 cm3, so the volume is the number of cubes.
- Add the numbers in the plan. Back row 3, 1; middle row 2, 2; front row 1, 1.3 + 1 + 2 + 2 + 1 + 1 = 10 cubes
- Write the volume. 1 cm cube = 1 cm3.Volume = 10 cm3
Check. (3 + 1) + (2 + 2) + (1 + 1) = 4 + 4 + 2 = 10. ✓
Method. The front elevation uses the tallest stack in each column.
- Find the tallest in each column. Left column: 3, 2, 1 → 3. Right column: 1, 2, 1 → 2.Heights 3 and 2
- Add. Each unit of height is one square.3 + 2 = 5
Check. Count the squares in the finished drawing: a column of 3 and a column of 2. ✓
Method. Find the volume of one cube, then multiply by the number of cubes.
- Volume of one cube. Volume = side × side × side.2 × 2 × 2 = 8 cm3
- Multiply by 5. There are five identical cubes.5 × 8 = 40 cm3
Check. The 5 cubes have volume 5 × 8 = 40; no cubes overlap. ✓
Watch out. The cubes have side 2 cm, not 1 cm: each cube is 8 cm3.
Method. Work out the side elevation yourself, then compare it with Mia’s drawing.
- Find the tallest stack in each row. The front row has stacks 3 and 1, so its tallest is 3. The back row has stacks 1 and 2, so its tallest is 2.Front row: 3. Back row: 2.
- Decide where each row goes. Looking from the right, the front is on the left of the view. So the correct side elevation has a column of 3 on the left and a column of 2 on the right.Correct: heights 3 then 2
- Compare. Mia’s drawing has heights 2 then 3: the rows are the wrong way round.Mia is not correct.
Check. She has reversed the rows. Her drawing would be correct for a view from the left. ✓
Watch out. Looking from the right, the front of the solid is on the left.
Method. The plan is one row, so there is only one stack in each column. Each stack is as tall as its column.
- Read the plan. A row of 3 squares means 3 stacks standing side by side, with nothing behind them.3 stacks
- Use the front elevation. Because there is only one stack per column, the heights are 2, 3 and 1.2 + 3 + 1 = 6 cubes
Check. 6 cubes in total; 2 + 3 + 1 = 6. ✓
Method. From the side you see one row only, and the tallest stack is the one that shows.
- Find the tallest stack. The stacks have heights 2, 3 and 1.Tallest = 3
- Describe the view. There is only one row, so the view is one column, 3 squares tall.A column of 3 squares (1 wide, 3 high)
Check. From the side the stack of 3 hides the stacks of 2 and 1. ✓
Method. The sloping side is the hypotenuse of a right-angled triangle. Draw the height down the middle to split the base in half.
- Find the half-base. The height meets the base in the middle.10 ÷ 2 = 5 cm
- Use Pythagoras: a2 + b2 = c2. The two short sides are 5 and 12.52 + 122 = 25 + 144 = 169
- Square root. Undo the squaring.√169 = 13 cm
Check. 5, 12, 13 is a well-known right-angled triangle. ✓
Watch out. Use half of the base (5 cm), not the whole base.
Method. For the greatest number, make every stack as tall as both views allow: the smaller of its column limit and its row limit.
- Write the limits. Columns (left to right): 1, 3, 2. Rows (front to back): 2, 3.
- Front row (limit 2). Each stack is the smaller of the column limit and 2.min(1,2), min(3,2), min(2,2) = 1, 2, 2: total 5
- Back row (limit 3). Each stack is the smaller of the column limit and 3.min(1,3), min(3,3), min(2,3) = 1, 3, 2: total 6
- Add the two rows. 5 + 6 = 11
Check. This arrangement still shows heights 1, 3, 2 from the front and 2, 3 from the side. ✓
Watch out. Check both views are still correct after you choose the heights.